On Android, AntennaPod is open-source and provides integrated search of iTunes, FYYD, and gpodder.net . You can also copy/paste a URL, for example if you were to find the URL for them and send it to them. Its interface takes a little getting used to, and search results don't always hit (which I view to be a matter of the sources searched, rather than the app). That said, it's a quite solid app. And free of crap-ware.
Seconded.
AntennaPod supports the two ways I want to add a podcast: search and by url. Haven't run into the problem of it not finding anything either (but if I did, I agree that it's not the app's fault).
Recommend it 100%.