Simplest answer – download itunes, subscribe to podcasts, open podcast folder within the itunes folder, drag and drop files to mp3 player.
Itunes haters will scream about this, but the fact is that itunes has about the best podcast listings around (many third party apps use itunes to get the podcast listing).
Most podcast apps are designed to work on smartphones nowadays. Podcasts use RSS for distribution, so you may be able to use an rss reader for that, although they are getting harder to find. Spotify will play podcasts, unknown if they will let you download them.