Open Source from the Watson Institute. Hosted by Christopher Lydon, this podcast is often focused on literature and the connections between literature and politics and society. His recent series on Pakistan was fascinating as was his recent interview with Nigerian/American writer (Open City). In the past, he's interviewed Helen Vendler on Emily Dickinson and Lila Azam Zanganeh on Nabakov.
I also like the Open Books podcast from the BBC. In this 30 minute weekly program, host Harriett Gilbert interviews two well-known (usually British) novelists or writers who talk about the book which has influenced them most. The guests and Harriett all read the book(s) and then discuss why it has been so important or influential. Recently they've discussed Jean Rhys, Thackeray, and women crime writers.
Of course, there's Writers and Company. No way one can mention literary podcasts without mentioning Eleanor Wachtel's highly influential program which has been available on podcast for many years now. If you only have time for one literary program a week, this is it.
I also like two not necessarily literary podcasts but programmes which often mention or discuss literature, namely NPR's Fresh Air (interviews with writers and hot political or social topics of the day: this programme truly is a key part of the "conversation" happening around culture and politics today and host Terri Gross has her finger on the pulse of contemporary life like almost no other personality) and Slate's Culture Gabfest (discussions of who's or what's in the news including writers, books, movies and personalities: again, a major player in shaping and determining the conversations we have around the water coolers in North America though the personalities and general banter of the hosts can get a bit irritating at times. The Endorsement title is a usual feature of their programme).
Finally, I can't forget about New Yorker Outloud, New Yorker's monthly Fiction podcast (a New Yorker writer reads a short story published in the magazine by a different writer), The Moth Podcast (stories told live on stage) and The Monocle Weekly (design, art, politics, social innovation, and, yes, occasionally literature).
Brûlé's weekly podcast - not just for rich people anymore (since the Recession)!