motor mounts... although that would be more evident on acceleration and when you get off the gas,
I second that motion.

You can do an in-driveway check to give you a general idea of whether or not the mounts are the problem, and which ones.
Find some level ground, it's easier. If it's an automagic, put it in park; if manual, use reverse or 1st
without engaging the parking brake. Open the decklid, now push/pull on your favorite body part in an attempt to rock the car and observe.
If the top of the motor reminds you of your grandma's favorite rocking chair, generally the first place to look is at the dogbone bushings. The bolt holes through the bushings should be snug & round, not elliptical.
If the bottom of the motor looks like it's trying to move about, you may have worn transmission, lower engine mount, or even the cradle bushings. 20-year-old rubber parts have a way of stretching & drying out.