Robert Fisher was right.

Two of the transmission mount bolts weren't properly tightened.

They weren't really loose, which is why when I was checking over everything
(I stared at that transmission mount at least 6 different times) but I guess
there was a little bit of wiggle room for the mount to pivot diagonally
across the two looser screws.  I think when I went to tighten them,  I
mistook resistance related to rust and the weight of the transmission as the
bolt cap tightening down all the way.

But now I have a problem.  One of the bolts, it seems, was made too tight.
Or maybe the threads were already trashed.

So now my noise is gone but one of the four tranny mount bolt holes has
stripped out threads.  Is a heli-coil the answer here?  I've never had to
deal with this kind of situation before.  Any help would be greatly(!!!)
appreciated.  I guess I have to tap threads, and then insert the heli-coil?

So, thanks to *everybody* that listened to my noise and gave me their
feedback.  It could have been any of the things people suggested, and the
encouragement to check over everything was exactly what I needed.

Thanks again!!