Can I make the Podcasts App play my unplayed podcasts in chronological order?
Unfortunately, no. 

The combined Unplayed Episodes playlist has no settings and defaults to newest first.

As of version 1.01, you can only sort individual subscriptions by oldest first. 

Even if you go through the trouble of changing each individual subscription preference to oldest first, Podcasts app will still keep the Unplayed Episodes playlist sorted to newest first.