Searches on the web proved to be futile. Numerous people had the same problem, but no one had a solution. I looked at the Garmin site support page, and the solutions they offered there didn't change a thing. I updated both my USB drivers and my version of Mapsource, and still the same error.
Now my blog has basically no readership at all, but I'm putting this up anyway on the offhand chance that someone with the same problem might stumble across this with a Google search. The key is in turning your GPS into a USB device (effectively a flash drive), and you do that as follows:
1. Hit Menu twice
2. Select Setup
3. Select Interface
4. Select USB Mass Storage at the bottom
Now you connect to your computer and upload the maps as before. The main difference is that you won't see "Garmin Device XXX" as an option. Instead, you'll be looking for the appropriate letter drive to upload the Maps.
Normally, a Garmin GPS will be identified as "Garmin XXXX" (76CSx in my case), and that's what I see when I normally connect it to transfer waypoints, tracks, and routes. But when you change it to be seen as a USB Mass Storage device, it just looks like any other generic flash drive, since that's what you've basically turned it into.