The Akkadian culture and language didn't go anywhere, the Assyrian and Babylonian languages are in fact dialects of Akkadian and their cultures were greatly influenced by that of the Akkadian Empire - Assyria and Babylon were simply successor states. As for the Sumerians, their language ceased to be commonly spoken by the time of Hammurabi, and their distinct cultural identity was subsequently forgotten over time as the area they once lived in became associated with and dominated by the Babylonians instead. More on reddit.com

Right, here goes. Some of this has been said by Hanguo but I wanted to massively expand on it.

So, first things first. Sumerian, Akkadian and Persian are all completely different languages. Language is obviously not the same as culture, but having a different language generally does mean you're not including yourself in someone else's culture and linguistic identity does play a major role in cultural identity. Sumerian is, so far as we know, a language isolate. Akkadian is an Afro-Asiatic language as was stated already, closely related to languages like ancient Egyptian, Arabic, Canaanite, Aramaic and Hebrew. Old Persian (as it would have been in this period) is an Indo-European language, relatively closely related to many Indian languages and several in Central Asia and also related to other big branches like the Celtic, Germanic and Romance languages.

The Sumerians did occupy much of the same territory that later Akkadian cultures came to occupy. The relationship between the two is somewhat complicated to explain. Suffice to say that Sumerian culture was already quite old by the time that Akkadian speakers moved into Mesopotamia, and they were already a complex society. Control over the region dovetailed between Sumerian speaking and Akkadian speaking cultures for some time, before Akkadian cultures definitively took over and Sumerian vanished as a vernacular language.

The reason that it's complicated is that despite everything, the Akkadians viewed both Sumerian culture and language with a high degree of prestige. There is a phenomenon in the ancient world in which one culture would call the gods of another using the names of their own gods, which is why you get Celtic gods being referred to as Zeus/Jupiter/Mercury/Apollo etc. In this case, the Sumerian gods were actually properly integrated into Akkadian religious beliefs as well and the mythology of the Sumerian culture added to that of the Akkadians themselves. Many of the ancient Sumerian deity names continued to be used alongside those of the Akkadian deities, and so the 'Mesopotamian pantheon' as it's often referred to is a real cosmopolitan mix of the deities of two different cultures that had somewhat fused with one another. Now, both Sumerian and Akkadian were written using cuneiform script, but the reason that this was possible was that cuneiform could theoretically be adopted for any language regardless of its relationship to the Sumerian language. In addition to this, many Sumerian words were directly adapted into Akkadian languages, and the Sumerian language itself remained a language of liturgy for millenia afterwards (though by around 600 BC there was an ongoing issue that there were not many who could read Sumerian and it required great scholarship to do so). Despite the fact that the languages are from very different sources, this does means that eventually there were similarities between the two.

Now, I need to expand on Akkadian culture a little bit. A 'unified' Akkadian culture, if it had ever existed, did not last long. Two distinct dialects (and I suspect there were more) of Akkadian began to emerge; the Babylonian dialect and the Assyrian dialect. The culture of these two regions of Mesopotamia began to shift away from one another as well; for example, the King of Assur (the major city of Assyria) was also the High Priest of the god Asshur, whereas the High Priest of Marduk in Babylon and the King of Babylon were two seperate individuals. So by the time we're into the Late bronze age, we're not really referring to any culture as Akkadian any more and it's instead just used to refer to a) texts written in older Akkadian dialects and b) the linguistic group that Babylonian and Assyrian belonged to, though perhaps also c) recognition that the two cultures were related but not identical.

Both regions became highly cosmopolitan, to the point where in Babylonia 'Babylonian' became a completely meaningless term given the dizzying array of peoples represented there. Both regions eventually became dominated by Aramaic, written with an alphabet, as their vernacular and therefore Akkadian too became a highly formal language that became rarer and rarer as the centuries wore on.

Then we come to the Persians, at last. The exact migration of Indo-European speaking groups is still not fully understand, but we are comfortable referring to the Persians of this period as being part of the Indo-Iranian world, and one of the Iranian speaking peoples. Their origin point is likely to lie somewhere in Central Asia. They were closely related to several other cultures, including the Medes, Parthians, Bactrians, Sogdianians and other Iranian speakers dotted around modern Iran and Central Asia. The region that they came to occupy, known as Fars today, was previously part of the territory of a very ancient and respected state called Elam. Assyrian accounts report the emergence of the Persians in the regions next to Elam, and their eventual military conquest of the Elamite state. So we can state that the Persians were in the Fars region and to some extent in Elam by at least the late 600s BC. They next feature prominently at the tail end of the Neo-Babylonian Empire, a state that emerged in the collapse of the Assyrian Empire (more properly the Neo-Assyrian Empire). Babylonian texts refer to Cyrus II (the Great) specifically, in fact, noting his military activities and his conquest of the Median Empire/Kingdom (I really don't feel the Median state deserves the moniker of Empire). This is when the Persians definitively establish themselves as the Empire, conquering Babylonia and the rest of Mesopotamia followed by many other nearby regions.

It's important to note, first of all, that the true capitals of the Persian Empire remained in Elam/Persia for the duration of the Empire. Babylon was an important centre but it was not the capital, Pasagardae, Persepolis, Susa and Ecbatana functioned as the main centres of the Empire. Elam/Persia remained the core of the Empire, although Babylon was another major integral part of the Empire as well. In addition, Persians did not settle in Mesopotamia, though Persian officials, satraps and garrisons were sent out to various parts of the Empire they did not practice any settlement policies. Mesopotamia, and Babylonia/Assyria, were conquered regions integrated into the Persian Empire and not considered part of Persia itself.

The administrative languages of Persia were mostly old Elamite and Aramaic. Elamite seems to mostly be confined to Elam and Persia, whereas administrative documents in Aramaic seem to occur in almost all regions of the Empire. They did not treat Akkadian as a diplomatic or prestige language as the Assyrians and Babylonians had done, nor did they adopt Mesopotamian deities (although they were very willing to honour them with sacrifices, temple re/building and dedications). Elements of Mesopotamian culture did make their way into Persian, for example in the style of palatial and temple architecture, but certainly nothing like the way that elements of Sumerian and Akkadian culture fused together. Persians were very much understood to be a different people, including from other Iranian speakers- only the satrapy of Persia itself was exempt from paying tax or tribute.

As a major Empire, the Achaemenid Persian Empire was surely the successor of both the Neo-Assyrian and Neo-Babylonian Empires, inheriting many structures and practices from both. But culturally, it was an entirely new influence on the region and I would argue that if it was a successor to any of the older cultures of the Near East it's to the Elamites (who they were not directly related to in the first place it's worth pointing out).

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The Akkadian language and Sumerian languages are not related though, and the writing system had to be heavily adapted to write Akkadian and was very clunky for that purpose. The modern Hebrew writing system is not derived from cuneiform, but from Phoenician letters (which ultimately have their roots in the Egyptian origin of writing, not the Mesopotamian one). There are still some Akkadian loanwords in modern Hebrew (e.g. אדריכל “architect” from an Akkadian word for “palace slave”), as well as some cognates since both are Semitic languages. More on reddit.com

Before I start, if anyone is interested in a detailed discussion of Sumerian phonology: Jagersma's Descriptive Grammar of Sumerian is open-access, it will tell you basically everything we know about the sounds of Sumerian. Our main evidence for the pronunciation of Sumerian are Sumerian loanwords into Akkadian, which was the Semitic language spoken in the same region, or loans from Akkadian into Sumerian. As Akkadian is Semitic, we are able to quite accurately reconstruct the phonology of the language by comparing it with other languages such as Hebrew, Aramaic and Arabic. Sumerian died out as a living language probably somewhere at the end of the 3rd millennium or early 2nd millennium BCE, so loan words from Sumerian into Akkadian from periods beyond this timeframe are not too useful. Thankfully there are a lot of loans into Akkadian from before the death of Sumerian, and these loans can tell us something about the pronunciation of certain phonemes. Take for example the set of velar stops: Sumerian had two, which we conventionally write as k and g. Akkadian on the other hand had three, which we write k, g and q. Through comparative Semitics we are pretty sure these three Akkadian stops were pronounced [k], [g] and [k'] (or glottalized). When we look at the loan words, a clear pattern emerges: words that include the Sumerian k and g are loaned into Akkadian as a k [k], and what we write as Akkadian k [k], g [g] and q [k'] are loaned into Sumerian as a g. This must mean that Sumerian k and g are closer to Akkadian k [k] than Akkadian g [g] and q [k'], and likewise Akkadian k [k], g [g] and q [k'] are closer to Sumerian g than Sumerian k. Our conclusion is therefore that Sumerian g was actually pronounced [k], and that Sumerian k was a sound close to [k] but not [g] or [k'], therefore we think it was probably [kʰ]. Hope this clears up the basics of our Sumerian phonological reconstruction! More on reddit.com