I haven't had that issue with my Hyundai, but I had a bad shake on my '92 Ford Escort(Mazda) about ten years ago. It would shake really bad only when I went over 40 MPH. Turned out that the all the bushings that the steering rack bolt to were gone and the steering rack, though attached, wasn't firmly mounted causing the shake. Removing that rack was not a fun job, it wasn't a cheap fix either. I had to buy a new rack if memory serves, but I did the job myself saving labor charges.

I would also check the engine and tranny mounts as well. I've had issues with that as well on the same car(as above), but my case only the engine would shake, not the whole car.