I worked this out, finally. The simplest way to do this turns out to be eye-rollingly simple. Just use the Zune edit feature to change the Genre to "Podcast". Open each mp3 in the editor by right-clicking on it, and clicking on Edit. The Genre dropdown may not have Podcast as one of the options, but it doesn't matter, just type "Podcast" in Genre and click OK. Done!
At this point you will have to copy or move the file(s) to the folder that Zune is recognizing as the Podcast folder and it will be recognized as such.
Wasn't that easy?
It turns out that there's another way to do this, but you can't do it in the Zune software. I won't explain it here, and it requires that you use a third-party tag editor, but you can find a description on the BlueSam Blog. A bit more complicated, but it has the additional benefit of making an mp3 appear as a podcast in iTunes (assuming naming the Genre "Podcast" doesn't work in iTunes, something I haven't checked -- I hate iTunes, that's why!).