Since I really do never downshift more than half way, I first removed the front derailleur, which a I could tell was not exactly on straight. The chain moved more smoothly, but it didn't really solve my problem.
Now if you do much searching for information on the internet, you have probably noticed that whatever you're looking for, 99% of what you'll find is just somebody trying to sell you crap. Google really ought to have a feature that lets you block commercial sites. But I'm happy to report that the subject of bicycles is an exception--you really can find sites that aren't pushing anything, and actually contain helpful information.
A little reading, and I resolved that some part of my drive train must just be worn out. I have been riding this bike a lot, for more than a few years now, and I am at least the third owner. One of the ways you can wear these things out is by stretching your chain. There's an easy way to check for this; each link is supposed be exactly 1/2 inch long, so if you put up a ruler against your chain, and line up a link at zero, and don't find that another link 24 down the line aligns perfectly with the other end of the ruler, then something's not right. Indeed, the 24th link away from the zero mark on my ruler was about 12&3/8" away.
I had a slightly rusty chain salvaged from some previous bike, and put it on. Triflow helps loosen these things up nicely. Now some of these DIY bike repair sites will caution you that if you replace your chain, you may have to replace all your gears to make sure they mesh properly. Apparently they wear together, and a new chain may not work happily with worn gear teeth (one useful tip is to try reversing your gears instead of just buying new ones).
I have had to make no such drastic overhaul, thankfully.
Here's my not-so-pimp ride, complete with CalTrain tag and "fight terror, ride a bike" sticker:
Lot's of folks have fancier bikes--sitting in their garage!