It can be done in O(E*V) without using the partition property. Start by deleting all edges with the same color on both vertexes, this can be done in O(E). In the modified graph G', every triangle is a 3-colored triangle. Finding the triangles in a graph:

for each edge e(u,v):
    for each vertex w:
        if e(v,w) and e(u,w) in G':
            add (u,v,w) to triangle list

If you keep adjacency list as well as adjacency matrix, you can improve the time of the inner loop by checking only w's in the adjacency list of v. In that case the complexity is O(E * max(deg(v)).