spec 3 problem??
There are only 3 parts holding the engine/trans in place... the 2 motor mounts, and the tranny mount. My first suspect is the tranny mount, of course - and it can be deceptive in looking OK, but being failed. Did you try jacking up on the tranny to see if the mount actually held the trans in place?
Beyond that, one of your motor mounts may be broken. Don't be afraid of polyurethane motor mounts (Thunder has the LS1 specific ones). They don't (IMHO) introduce any more vibration than rubber, but are a much better mount. I run poly in my 98Z and my 79Z. The only painful aspect there is the actual replacement process.

Good luck!

