I have a 1988 GTU, and I just had the clutch and flywheel replaced with the Bonez street comp clutch and flywheel. Prior to this, the clutch had worn completely smooth, and was sitting around for a while. However, before that, aside from the worn clutch, the car was fine. The work was done by the local mazdaspeed dealership.
So, when accelerating from a stop, right off the line, if I don't dump the clutch with at least 1/2-3/4 throttle, the rear end starts to vibrate violently (not enough to make it difficult to drive, but enough to be cause of concern to me), but this only lasts a couple seconds, then it smooths out. I tried accelerating from a stop in second, but this still happens sometimes, though not as violently.
My friend, the co-owner of the car, thought it was one wheel slipping and the other getting traction, but I don't think it's that, as it doesn't feel similar to when the car burns out.
I would say the closest thing I've felt to this before is wheel hop, but I've only experienced wheel hop in a fwd car, so I don't know if this is that, or something else entirely. The suspension isn't at 100%, and the alignment isn't perfect (it had a 4-wheel alignment done, and the tech said that he got it as good as he could, but there was some stuff he wasn't able to fix entirely), and there's no rear strut tower bar, so I dunno if this might be the problem.
From a search of the forum, the problem sounds most like the (1 or 2 sentence) descriptions I've seen of a bad front u-joint. Does my problem sound like a description of this to someone who knows what that is like?
Basically, what on earth is this that's happening?

 

It does sound like a U-joint. In which you should check out Rockforddriveline.com part number 430-10A you need two.

You'll need a 4lbs hammer and a punch (I used a old cheap socket), a drimmel with the cut off wheels and the little sanders. its an easy job once you get the drive shaft out.

 

Is the differential an LSD? Does it feel/sound kind of like chatter?

 

yes, it's the GTU LSD. It feels like something is vibrating the back end of the car. Not really fast enough to be what I'd call a "chatter", though. maybe 5 - 10Hz, but it shakes the back a fair bit for a few seconds, then dissapears. As I said, it does feel a little like wheel hop.

 

Reason I asked is because I had something similar to this from my LSD and a differential additive cleared it up. I would call mine more of a "chatter" though, but some of your symptoms sounded the same.