National Public Radio podcasts the weekly "Selected Shorts" radio show. It's pretty tightly constrained into a one-hour format because of the original medium, but is usually fun. I think it updates on Sunday or Monday, but I'm not sure.
On the topic of public radio, This American Life is available as a free podcast. It's not fiction, but has stories interesting enough that they might as well be. Host Ira Glass from Chicago Public Radio presents one to three stories on a given theme. If this week's episode, The Super, is still available, you simply HAVE to listen to it.
The New Yorker magazine also has a fiction podcast. This is pretty nice, even if a bit on the highbrow/semi-pretentious side. A current writer chooses a story from the New Yorker's fiction archives and introduces the author. After you hear the story there's a little discussion and analysis of it at the end, by the same writer who did the intro. It's nice to see what other writers like to read and the podcast is well-produced. It is a little short, though, and only updates monthly.
There's also something called the Classic Tales podcast that highlights older stories. It's well-produced, but your enjoyment is dependent on how you feel about a certain slightly archaic style of writing.
Steve has mentioned The Drabblecast. It updates weekly and has very short (usually 2000 word or less) stories that usually have twist endings. The production values are top-notch and the quality of the reading is some of the best I've heard. It is marred by what I consider slightly sophomoric introductions with little to do with the actual fiction, but if you can get past those it's worth a listen.
Hope this helps. Enjoy.
