The BBC’s iPlayer Radio apps are great for catching up on radio shows and now they have implemented podcast downloads. So you can download and listen to a BBC podcast offline without a network connection. You can queue downloads and set it only download on WiFi if you want to avoid data charges. The Android version of the app displays download progress as well as having quick play option so you can play podcasts from the notifications bar.
The download features only work with BBC podcasts and not all BBC radio programmes, so you can get show like BBC Friday Night Comedy but not shows like the Radio 1 breakfast show due to rights issues (but that could change with a recent BBC Trust ruling)