Oh boy, you asked for it.
We Hate Movies - Have to put this first since I created /r/wehatemovies and am a mod there. Fantastic show taking down terrible movies via great improv comedy. Very relaxed and super funny.
I Was There Too - One of my new favorite podcasts. Each episode the host interviews actors who had small parts in major films. In one of the recent episodes he interviewed several of the bus passengers in Speed.
Star Wars Minute - Two smart nerds analyze each of the original Star Wars films one minute at a time. Ultra in-depth but thankfully they aren't complete Star Wars nerds and make fun of a lot of the stupid stuff, especially all the extended universe madness.
James Bonding - Two very funny comedians tackle each James Bond film with a fellow funny guest. Very good, just avoid the Goldfinger episode as your first episode.
Denzel Washington is the Greatest Actor of All Time - Two very funny comedians tackle every Denzel Washington movie (see a pattern here?) but use that as a platform to talk about race in movies. A bit preachy but I love it and I'm very white!
The Film Programme - My preferred British review show. Kermode and Mayo is more cultish and worth checking out but it's not my go to. This is a quick 30 minutes but usually fits in 4 or 5 segments including reviews and retrospectives.
The Business - Great show I wish was a lot longer. Focusing entirely on the business side of Hollywood, the show starts with a quick rundown of the latest business news. The second part is an interview with a major Hollywood figure about their work in relation to the business side.
The Treatment - Wonderful interview show with probably the smartest interviewer out there, Elvis Mitchell. Guy asks really strange but smart questions. Real good if you like the NPR style interviews.
How Did This Get Made? - There's a shit ton of bad movie review podcasts out there and this is the only other one I like. This podcast is loud and is more about shouting and big moments. Not every episode hits but with the right combo of guest and film it's great.
Filmspotting - The mainstay movie review podcast and with good reason. The two hosts are perfectly complementary, the segments are interactive and fun, and the top 5 lists that end each show are clever ways to garner recommendations.
Filmspotting: Streaming Video Unit - Spin-off of Filmspotting focusing on films available for streaming or rent. Each week they tell you what's new on streaming and review a film or TV show chosen by the audience.
Hollywood Handbook - Great satrical podcast parodying douchey insider types. Lots of idioicy but very clever and just infuriating enough.
Scriptnotes - Pretty good screenwriting podcast with two hosts with real experience. The hosts aren't my favorite but it's a good show.
The Canon - Two critics each week debate whether or not a famous film should be included in the canon of all time great movies. The audience then gets to decide if the film makes it in.