There is some sort of software switch in iOS 6. If the podcasts app is installed on the phone, then the default behavior of the Music app changes, and it will no longer display podcasts that are on the phone. You can undo this behavior by deleting the podcasts app, and then restarting your phone once or twice. After that, you should be able to see and play podcasts from within the Music app again (like you could in iOS 5).
Aftter doing this, the music app will also let you create on-the-fly playlists to which you can add podcasts. I do this to queue up multiple podcasts for long commutes, so I don't have to fiddle with the phone while driving.