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I have a 2010 Civic EX, and often I park on my sloped driveway with the nose of the car lower than the tail of the car. When idling on this slope, the car shakes pretty violently; even worse when the AC is flipped on.

Does not occur on flat ground, although there is still a considerable amount of engine rumble in the cabin.

Any clue what could cause this? My gut reaction says broken motor mount. What else could it be, or is this typical for Civics?

 
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Ya pretty sure it is a motor mount. Probably the front tranni mount, the front tranni mounts are known to go bad alot. You can get some energy suspensionb motor mounts for like $20. All you do is slip them in the original mounts. This is actually a very good thing to do. It helps engine response and stops wheel hopping and a couple other things. Though it does make your car vibrate a little but not much.