Agreed with Chapo Trap House (and surprised it wasn't mentioned), have never tried Cum Town. Katie Halper I've tried a little bit but it hasn't stuck.
Michael & Us is terrific and I have many to catch up on. The mission statement was to cover all of Michael Moore's movies and mostly trash them (from the left politically and from a film crit perspective), but they've covered many other politically-related movies after finishing that. I linked to the episode on Christopher Nolan and The Dark Knight Rises. Most recently they started a miniseries on South Park.
The Important Cinema Club shares one of those two hosts and is a more general film podcast. I linked to the most recent one, on Paul Verhoeven (Total Recall, Robocop, Starship Troopers, etc.), and how he was completely overlooked as a serious filmmaker for years. Everybody missed the boat completely on Showgirls.