We checked the UDP issue already (tho H.323 uses TCP there), and nothing was getting through there either.
The problem was solved with an upgrade to the OS on the Cisco router--something to do with
this. It turned out that 1720 was the ONLY port that exhibited this behavior.
I loved the bit about the "hidden command." Reminds me of double secret probation. (I'm also a little miffed that I found this out myself while tech support droids at three different companies were insisting they had never heard of anything like this and I must have screwed something up. They couldn't tell me what, but they were all sure it was on my end.)