Figured it out:
Can I use the Apple Podcast app to download my Premiere Radio Networks subscription podcast?
Unfortunately, the new Apple Podcast app is not compatible with paid/authenticated podcasts such as ours. Thus, the app cannot be used to automatically download new episodes through the Apple Podcast service. Episodes of our podcasts will continue to have to be downloaded to iTunes on your desktop or laptop computer, then synced to your iPhone or iPod.
The Podcast app can be used to listen to your synced episodes. Be aware, however, that once you have synced your episodes from iTunes to the app, you may begin to receive alerts from the app asking you to log in to download the most recent episodes. This is despite the fact that, as stated above, the app cannot be used to update in this fashion.