I'm not sure what I expected to find, exactly. I guess I just have a weird idea that if such a thing can be imagined by my mind, it must exist somewhere on the internet. As it turns out, I was only able to find a single dino-centric podcast. And it only had one episode, devoted entirely to Leonardo. It was called This Week in Dinosaurs, and I haven't been able to tell what happened to it. A post on the forum mentions server problems, but also says that a new episode would be up within a week. That was last November.
While it's not surprising that someone made a podcast and it only lasted one episode, the production seems to indicate that the host, someone named Eric, had higher aspirations. His delivery as he opens the show sounds inspired by Ira Glass, and he had no less than Robert Bakker as his first guest. He has an annoying habit of interjecting little "okays" as his guests speak (he tensds to let them ramble), but it's not nearly as annoying as the incessant interruptions of Science Friday's Ira Flatow. The music, which is an appropriately science-y bit of electronica, strangely plays throughout all 35 minutes of the episode. These are things that I imagine would have been ironed out with experience.
Also: Threadless is featuring a new dinosaur tee called "The USA is Dino-mite!" I'm not sure if it's advocating gay rights, or if the T. rex is just wearing fancy pants.