Isometric might tackle these problems -- you wouldn't need 3D models, you could use art from standard paper minis, etc. But I'm sure it has it's own problems, like not being able to click on figures "behind" or northwest of other figures. Or the extra details might be lost when shrunk to a playable size.
Anyway, while I like the extra features of the side-view, they're lost when shrunk to grid size. Maybe I'll just save the side-view for the chat box and go with round tokens, but bigger round tokens for large creatures, and avoid the top-view.

Evan said:
One thing to bear in mind is the interaction of your token and the amount of grid it should consume (assuming D&D 5e): http://dnd.wizards.com/products/tabletop/dm-basic-rules
A large creature should cover 2x2 squares on a 5ft grid. If your creatures are elongated on the grid it might not accurately represent the amount of space they control. Perhaps you could use the shadow to make the token stretch out appropriately in the x-axis?
I agree there is a tactile and psychological effect that comes with big minis, but that's also a facet of them existing in 3d space. Perhaps one day will we get a 3d VTT with Roll20 (or there are alternatives if you look online) and in that case you can utilise the height to provide the feeling you want.
Anecdotally: I often struggle when I'm describing a monster's height as the stat blocks rarely explain how "tall" a Large or Huge creature is ;) I think there's some charts out there that try to express this but they are not exhaustive. I know it might be asking a lot, but it would be cool if future editions had some basic height/weight stats included.