Here is a video of how to check universal joints, and what you will see when one is bad (this one is really bad, any play is unacceptable). Your Z has 6 universal joints.
Check them all.
The front differential mount is two pieces of metal joined with rubber. One of the metal pieces is bolted to the differential, the other to the cross member. If both parts are bolted securely, then it is possible that the rubber has torn away from the metal. To check for this use a prybar too lift the nose of the differential, and observe if this is the case.
The transmission mount, and the engine mounts are of similar design, and are checked the same way.
Check them.
Also, check the exhaust. Grab the muffler outlet and see if it wiggles, observe how much, and if you hear any part of the exhaust hitting anything. A little wiggle is OK, a lot, not so much. Examine all of the mounts, repair or replace any broken parts. Bailing wire or coathangers are not acceptable repair materials.