Okay, so instead of just big lists, which I like, but aren't really useful for knowing which are anyone's favourite, how about we all recommend just ONE podcast? Not your favourite podcast, but the one you are most excited or interested to share.
I'm going to go with the Pessimists Archive Podcast. There are only three episodes so far (monthly podcast, I think), but they've all been good. It looks at technology that, when it was first released, there were many predictions that it would be the cause of the end of society.
The first episode was about the Walkman. The second was about when "the good old days" were, when we say things used to be better. The third was about recorded music itself. I like the reporting, the stories, and the balance of the viewpoint.
Here's the website: http://pessimists.co