How would you guys do this?
 

I want to animate flat cartoon like faces on the front of these heads but what would be the best way to do it.

First i was thinking animated texture but not only would that take ages it'd be a pain in the butt to get the timing right. So then i thought how about making some flat 2d shapes for eyes and mouth then push them up right against the face and turn cast shadows off, then i could just use blend shapes on them. But i don't know so i thought i'd ask before wasting time.

I think flat shapes is a good way to go. But I think the best thing would be to use the same box head face that I remember seeing in "Stop Staring" by Jason Osipa. I think this method would work really well with what you are tring to accomplish.