The fix was to open the podcast settings dialog and change the "episodes to keep" option. The problem is that in the latest iTunes, it's not obvious where to find that dialog box. It's not in the main preferences dialog. Instead find it by going to the podcasts view. Then look at the bottom of the window. You'll find "Unsubscribe" and "Settings..." buttons. Click "Settings...", then change the "Settings for:" field to "Podcast Defaults". Set "Episodes to keep:" to "All unplayed episodes".