A sampling of websites suggested that many people consider this behavior to be normal and unchangeable. It seems that it is neither. According to this item from the Apple discussions board, the solution is to throw out (or, better, archive!) your iTunes Library file from ~/Music/iTunes/, then restart iTunes and rebuild the library from scratch. Specifically, use File / Add to Library... and select your iTunes Music folder, then sit for a few minutes while it chugs away. Best to be connected to the Internet so that iTunes can re-import album art.
Note that this will wipe out most or all of your playlists. But I tried it, and it solved my problem -- my podcasts now play in the order that I have them loaded.

