Four days. Four days, fellow Google Reader pilgrims: that's how long you've got until Google turns its back on the RSS service forever. Apps that used to rely on Google Reader as a backend have switched to alternatives, usually Feedly's new and almost identical backend API. Popular podcast manager/player BeyondPod is the latest to do so, but in order to try it out, you'll have to leave the comforting confines of the Google Play Store for the treacherous waters of a non-Market Beta.
You can download the latest BeyondPod beta (version 3.1.31) for smartphones here. The tablet version is a separate app, and the beta (with Feedly syncing) is downloadable here. You can add new podcast feeds and play them within the app, but in order to retrieve your old feeds all at once, you'll need to open a Feedly account and import your Google Reader data, if you haven't done so already. It's not hard - just head to Feedly.com, follow the instructions, then launch BeyondPod on your phone or tablet and select Add feed>Import feed and tap the refresh button. Step-by-step instructions can be found here.
From this point forward, BeyondPod will support syncing to Feedly only, and there's no indication that the app or service will be adding support for any of the other competing RSS readers that have cropped up. Expect to see these features added to the Google Play versions of BeyondPod eventually, and hopefully before July 1st.