I have this issue on my iPhone, iPad, and the iTunes app on my computer, which leads me to believe that it's not just some bug in the app. This happens for all the podcasts I have. The weird thing is that it actually knows which episodes are played, and indicates this with a little gray circle next to each unplayed episode, but for some reason the "unplayed" menu only shows one episode.
1 Answer
What I ended up doing was pressing "Edit" at the top left of the screen, then selecting all the episodes I haven't listened to yet (yeah it takes a minute), then pressing "Mark", then "Mark as unplayed". I don't have an answer for "why" though.