WTF with Marc Maron - Bi-weekly independent podcast. Maron is a grandaddy of the NY alternative comedy scene who never really made it big, until he sat down in his garage and started talking to his peers. It's a podcast about funny people that is often times miles from being funny, but that beautifully captures the essence of who Maron and his guests truly are. First podcast I download every week.
This American Life - Weekly podcast, originally from WBEZ Chicago. Ira Glass and his gang of storytelling nerds choose a theme and compile a set of interviews and fictional or non-fictional short stories to (loosely) explore that theme. I've been introduced to a lot of great writers through that show. On the occasions when the show is dedicated to a single story, the show's take is often refreshingly objective and inquisitive. Also, for a weekly podcast, their themes are often surprisingly ahead of the curb.
RadioLab - Weekly podcast, from WNYC. This is like This America Life, but exclusively dedicated to science.
Kermode and Mayo's Film Review - Weekly podcast, BBC Radio 5. The good doctors discuss the UK box office top 10 and the films released in the UK that week.
The Moth - Independent weekly podcast. The Moth is an organization putting on a series of storytelling shows across the US (they started in NY), inviting celebrities and regular folks alike to tell a story on stage. Each week, they release one or two stories from their huge vault of stories. They can be hit and miss, but there are a lot more hits than misses.
Risk - Independent (bi-)weekly podcast. In many ways, this is like The Moth but on drugs and with a lot more sex. Kevin Allison is an ex-member of The State, an unfunny US-based comedy group that made it big in the 90's. He's not exactly funny, but he's a good storyteller and he's created, participated and hosted this show for the last couple of years, based on the the theme of telling stories "you never thought you'd dare to share". Like The Moth, it can be hit-or-miss because it's based on people telling stories, and some will be much more compelling than others, but overall it's well worth it. Some truly screwed up stories in there.