Right, I've been playing X for ~5 hours now, and I'm pretty sure I've got an optimal strategy for how to get a pokemon fully amie-d out sussed out.

The way amie works is that each mini-game you play grants you cakes based on how well you do -- higher difficulty levels grant better cakes which give more affection in the form of hearts -- and grants a few hearts that can be gained by rubbing your pokemon. This is, as far as I can tell, the optimal way to use this system:

1. Choose a pokemon, enter amie.

2. Play "Head It" until you unlock the "unlimited" mode. (With some skill, this is easy, and Head it is the easiest and most reliable mini game).

3. Play 2 games of head it, aiming for 3 stars or above both times (Get a combo of ~50, or two of 30 -- you're basically aiming for 450+ points).

4. Quit playing, rub your pokemon until it stops giving you hearts. You should get 10 hearts from this.

5. Feed it full of the high quality cake from unlimited mode -- the swirly topping'd cakes give 5 hearts each.

6. Every X hearts give an "Affection level", which is signified by a HUGE outpouring of hearts from your pokemon.

7. Repeat 3-5 until you're at maximum amie bonding. This takes me ~35 minutes per pokemon.

HOT TIP: If you are getting four stars or above consistently, you can intentionally lose a game or two -- this reduces fullness and grants you 5 "rubbing hearts" the same as playing a good match.