"Is there some sort of podcast RSS-like program I can use to queue them up and then forget about?"
Yes. Podcasts are one of the original purposes of RSS. Podcast feeds are RSS feeds with audio or video files linked in the articles. You can find programs which do the refreshing and downloading on most platforms. I suggest Antennapod for Android (check FDroid for it) as a nice free start option or the included one for IOS. If you don't end up liking those apps but still like podcasts, there are about two hundred alternatives to be found in the app stores. There are desktop clients too.
"And would it work on my bedside clock-radio?"
That depends what your clock radio runs, but if it has an internet connection, quite possibly. Or if it can act as a speaker for something else, you could run it on that.
You will find a bunch of podcasts exist out there, and a great many are rudimentary or uninteresting, just like everything else on the internet. Still, I've found a lot of them which I enjoy, especially while I'm doing other things where I can't be reading The Register.