The TP Podcast includes some stories (& our TP website Literature section here has more on their authors).

Is Full of Crap: 100 word stories

Miette's Bedtime Story Podcast Short stories by famous writers

Nature Stories Podcast Weekly podcasts on the natural world, curated by Atlantic Public Media, sponsored by The Nature Conservancy and the Public Radio Exchange

PseudoPod - a Horror Fiction Podcast
Escape Pod - The SciFi Podcast Magazine!