I found a simple Powershell script on another question/answer site that works. I made some minor changes (added comment block explaining how to use it, added parameters so the necessary info can be provided up front).

Here it is:

<#
    Send-WOL.ps1
    Sends "magic" Wake on LAN packet to attempt to wake the target computer
    It is necessary to specify another computer on the same subnet as the target since the packet
    is only broadcast to the subnet the target is on
    Parameters: Target (required) : MAC address of target (xx:xx:xx:xx:xx:xx)
                RemoteIP (required): IP address of another computer on the same subnet as target
#>

# Setup params
Param(
    [Parameter(Mandatory=$true)][String]$Target,
    [Parameter(Mandatory=$true)][String]$RemoteIP
)

Write-Verbose -Message "Converting MAC address $($Target) to a byte array" -Verbose
$MacByteArray = $Target -split "[:-]" | ForEach-Object { [Byte] "0x$_"}
[Byte[]] $MagicPacket = (,0xFF * 6) + ($MacByteArray * 16)
Write-Verbose -Message "Broadcasting Wake-on-LAN packet to target." -Verbose
Invoke-Command -ComputerName $RemoteIP -ScriptBlock {
    $UdpClient = New-Object System.Net.Sockets.UdpClient
    $UdpClient.Connect(([System.Net.IPAddress]::Broadcast),7)
    $UdpClient.Send($using:MagicPacket,$using:MagicPacket.Length)
    $UdpClient.Close()
}

Wake-On-LAN uses a "magic packet" sent at Layer 2 of the OSI model. Routers need information that is contained at Layer 3 of the OSI model in order to route packets. Switches on the other had are Layer 2 devices, which is why it works when you are just going through a switch. In short, you need to have IP information in order to cross a router, WOL only has MAC address information, so it is not possible to cross the router.

Further, the computer that you are wanting to wake is not turned on, and therefore does not have an IP address, so it would be impossible to send a packet and have it cross the router and reach your computer short of configuring your router to statically route the packet to your NIC's MAC (see Wake-on-Internet for additional options).