Phone starts playing podcast for no apparent reason
I was having trouble with the original pre-installed "Music" app on my Droid X randomly playing music even though I never used the app.  I tried factory resets, updates, cache wipes, data clears, everything I could think of.  The only fix I found was to root the phone and use Titanium Backup to "Freeze" the Music app so that it no longer functioned.

It's possible that Music has latched on to one of your podcasts the same way that mine latched on to a song.

Note: this is not the new Google Music service and app that uses the cloud.  This was the original  Google Music app that came on the android phone.  Shame on Google for re-using the same name for two different apps, and both buggy.