We have a 2000 saturn (L series) with 175000 miles. We put 4 new (inexpensive) tires on last fall. I don't know if it's coincidence or not but ever since the new tires, the car started shaking violently while driving between 50 and 55 mph. Now, as soon as you start, it shakes violently. If you let off the gas, it quits immediately. Also, when you first start driving the car, it doesn't do the shaking until you have gone about 3 miles. We had the tires spin balanced, checked the motor mounts, put fuel cleaner additive in. We also rotated the new tires and that helped for about a week then it started shaking again. The shaking is not just a steering wheel shake but an all over violent car shaking. Could it be a cv? The fuel filter (we don't know where to find it)? Any other suggestions?

You have to better describe the shaking. Is it a side-to-side shake, like the cords in the tire may have shifted/broken? Is it a forward-and-backward shake, like the vehicle is losing power and regaining it?

I think you will find a blister on at least one tyre, probably rear. Jack each wheel up ( one at a time ) look at each tread all the way round ( put a chalk mark on tyre tread, this way you will know when you have seen a full revolution ) look for a bulge. Check both sides of tyres also.

Sounds like a bad inner CV axle. Give it a twist back and forth to check excessive backlash.

Think of the outer CV joint like your wrist. You can twist it round and round.
Now think of the inner CV joint, what does it do? Just goes along for the ride. But it does deal with a lot of torque.
Now, if the inner wears out just like the outer can, what will this result in?
This is an ASE drivetrain test question. Guess Mr Sticky Sam better brush up on basic automotive 101!!