As best I can tell the "Unplayed" panel is useless. You're stuck with playing through all your podcasts from the newest onwards (what on earth do Apple's software engineers think we do with podcasts?!). Plus all the description is a massive bloat in the interface.
More useful: "My Podcasts". When you first check that pane it looks ridiculous - a long list of all podcasts you've ever used, with most of mine at least having 0 episodes. But in the upper left corner you can press the "+" button to add a podcast or, more importantly, a station. You can then make a station that will include the podcasts you want: in my case, it's all of them, in the order I set them to long ago. There seems to be some moderately fine-grained control in the settings, although I haven't played with it yet. You can also set up a bunch of different stations. After that, when you choose the "My Podcasts" tab, any stations you've made will be listed at the top of the screen.
Unfortunately it seems like the app always opens to "Unplayed" rather than "My Podcasts", but I guess like with the Music app, I'm going to have to get used to clicking through a few pages before I get to the list of playlists that I want.