1. The Memory Palace: Let Nick Dimeo whisper in your ear with his sexy voice about strange moments in history. He knows how to tell a mean story.
2. The New Yorker Fiction Podcast: A current New Yorker fiction author reads a story from the New Yorker archives. Then the author and Deborah Treisman (the New Yorker's fiction editor) discuss the story. Excellent literary fiction accompanied by smart story analysis.
3. Radio Lab: Narrative based and easy to understand science reporting at its best. They take hard to understand science concepts and make them compelling. This is a delicious podcast.
4. Toasted Cake: It's new, it's being run by the mighty Tina Connelly who has great taste in fiction, and it's a great choice when you want something short.
5. The Moth: Slam poetry is to poetry as The Moth is to short stories. People bring their A game to this live storytelling podcast. Often outrageous.
6. Africa Past and Present: Stumbled upon this one when I was doing research on South Africa. It's academic, historical, and fascinating. I don't know about you all, but I don't know nearly enough about Africa and this podcast often expands my mind about humanity.
7. This American Life: It's been around forever and has rabid followers. Really good at building suspense and narrative tension in various ways.
8. Tor.com: For my money, this is where the most interesting speculative long-ish fiction is being published.
9. Escape Pod and Pod Castle: These sister casts are impeccably produced and are a great way to keep up with the SF short fiction world, as most the best stories tend to show up over here eventually.
10. Cast of Wonders: Does the world need more young adult speculative fiction in it? Yes. How can you even ask such a question?