I am familiar with WP 7.5 and Zune podcasts. In that system it is possible subscribe to a podcast by entering a URL into Zune. Once a podcast episode file is sync'd to the phone, you can set up the phone to subscribe to the podcast directly on the phone, over wifi, without using Zune.
Everyone knows that Microsoft failed to support this use case in WP 8.0. However I think there may be a way to do it in the WP 8.1 native Podcasts app.
My theory is based on a note displayed in the Podcasts app. To see it, go to the Favorites page, tap More (...), then tap Edit. You see a list of all "favorites", with a checkbox for each item. This appears to be a list of podcasts that the phone is subscribing to, i.e. will download episodes in the background. At the bottom of the list is a curious note:
You can also make your own Favorites Tiles. Just tap and hold a series in your collection, then tap Add to favorites.
This suggests that if you could put a podcast episode into your collection, with metadata about the series attached to it, then you could "Add to favorites" and thus begin downloading episodes in the background on the phone.
But how do you put a podcast episode into your collection? And once placed there, does it really work the way I think that it would?
I tried to copy an episode (which was downloaded using Zune) from my PC using Windows Explorer, into the \Phone\Music\Podcasts directory. But when I did this, all the metadata about the podcast series was lost.