I have a 1971 Convertible with a 4-Speed and the 270 HP engine.  There are no driveling modifications that I am aware of.  (Eng, Trans, Diff all numbers matching.  My Problem:  When driving the car it occasionally makes a sound directly under the transmission tunnel that sounds like the driveshaft or fron u-joint may be making slight contact with the body. This is intermittent and occurs during regular driving (at legal speed, in-town tppe driving)  I have had the body mount doughnuts replaced by a reputable body shop.

[QUOTE=red70-oneconv] [/QUOTE] Yes, the engine mounts or transmission mount can be broken. Also make sure that the transmission crossmember is secure at both sides.

MOUNTS MAY BE WEAK AND ALLOWING THE ENGINE TO TORQUE OVER TO MUCH. I WOULD LOOK UNDER CAR FOR ANY SIGNS OF IT HITTING (MARKS). SOMETIMES YOU CANT TELL IF MOUNTS ARE BAD JUST BY LOOKING.

Check your park brake cable...if it's loose, it can contact the driveline while your driving...just an added idea. Also, check the foil-type insulation that fits over the top of the trans. It can become loose over time, and slide back, contacting the front u-joint.
Normally, if the mounts, engine or trans, are bad, it will create a vibration issue. For sure, check to see if anything is hitting the body...don't need no driveshaft sticking up thru the floor...

[QUOTE=red70-oneconv]I have had the body mount doughnuts replaced by a reputable body shop.[/QUOTE]

Did you have this issue before the mounts were done?

Yes.  I had the problem before the body mounts were replaced.  I had them replaced due to advice from a body shop that some mounts could be missing or broken and this could be causing the problem.  Thanks

Another thought.  If anyone has ever jacked up the car from the center of the crossmember, it could have bent it up.  Not the correct place to raise the car but it has been done.  Solution-put it on a frame rack and bring the crossmember back to the place where it should be.  A clue.  Put a straight edge on the crossmember with equal blocks on each side about 1" thick so the straight edge clears the exhaust holes.  If you don't need the 1" blocks, the crossmember is bent up in the center.  A very common problem.  The crossmember should be straight with the exhaust holes slightly below the straight edge.  You will not find this measurement in the frame books but the problem described happens more often than it should.  The crossmember is not strong enough to lift the car without bending it up.  Good luck.