first thing to check is the passenger motor mount. It's located between the engine and the coolant resevoir and is easily visible..... big silver and black thing connecting the engine to the car. The stock mounts tend to not fare well in the long run. You're looking for signs that the oil inside the mount has leaked out (dark stain around the mount or below it), signs of drying/cracking to the rubber, or of complete breakage. Even if it's completely failed the engine isn't going to fall out of the car, but it will have several more inches of travel and can thunk up and down.
If it is your PMM you can replace it for about $100 for the OEM one.... not sure how much aftermarket ones go for. You just need a floor jack and some basic tools. Two bolts, two nuts and swap it out.
Beyond that check out your suspension components. Shocks/struts should be replaced every 50-75k miles. As was noted above also check the end links/bushings.
All that said I'm betting on the mount.