ALL OF THEM! D: I am such a podcast-addict, it's ridiculous.
I listen to a bunch of stuff, but I'm mostly into the narrative-format history-themed ones - I mean the ones that devote their entire podcast series to one area of history, and take however many episodes they need to do it. It lets me kind of sink into the subject and really learn, and unlike those podcasts that skip to a new subject with every episode, I don't have to be introduced to something new every time.
So, here goes!
History of Rome by Mike Duncan - how many Roman emperors can you name? If one of them is Elgabalus, the cross-dressing bisexual Syrian teenager who believed himself to be literally a god, I will be deeply impressed. If you haven't heard of him, 179 episodes of Mike Duncan's chronicle of Rome awaits you! Follow one of the world's greatest empire from rural obscurity to center of the known world to absolute ruin.
The History of Byzantium by Robin Pierson. Picks up where the History of Rome left off in the late 400s and follows the Eastern Roman empire and its crumbling existence, with the intention of going all the way to the 1400s and the capitulation to the Turks. Dives really deep into stuff like early Christian church-schisms, daily life in Constantinople, political structures, ethnic conflicts, etc., etc.
The British History Podcast by Jamie Jeffers. One day, Jaime Jeffers woke up with the idea that he was going to do a podcast, detailing the history of Britain from its earliest Ice Age-origins up until World War 2. It's been 4 years and 173 episodes, and he's only just getting to the Vikings. If you've ever asked yourself the question "What did British people EAT in the year 672?", this is the podcast for you. And even if you haven't! Jaime gets really deep into every aspect of British history, and he's doing a champion job of it too.
History of the Crusades by Sharyn Eastaugh. Why is the Middle East such a mess today, and what did the Holy Roman Emperor have to do with it? Listen to this podcast, and you'll find out! Having wrapped up her chronicle of the Crusades as they are traditionally considered, Sharyn Eastaugh has moved on to the more obscure ones, like the crusade against the Katars, in which the Catholic church fought other Christians in Southern France. Be warned, though; she does NOT flinch away from the goriest aspects of the crusades. There is a LOT of blood and violence and mutilation.
Revolutions by Mike Duncan. Having wrapped up his Herculean task of telling us the history of Rome, Mike Duncan has moved on to another topic - revolutions. He's gotten through the English Revolution (which brought Oliver Cromwell into the historical limelight) and the American Revolution, and is drawing to the close of his section on the French Revolution. Next up - Haiti!
The Egyptian History Podcast by Dominic Perry. I'm only a few episodes into this one, but it's nice to listen to a podcast on Egypt written and narrated by an actual Egyptologist who is intent on clearing away all the mysticism and myth and outright lies surrounding the history of Egypt. Lots of talk about archeological sites, the unreliability of historical narratives and pottery in the first few episodes!
Stuff you Missed in History Class - if you, unlike me, don't WANT to spend 100+ episodes learning about the same thing, this podcast might be for you! 1-2 episodes on each subject, getting into everything from sad royal childhoods to 'orrible murders and the history of peanut butter.
Also, like @rainbowolfe I listen to Welcome to Nightvale. It's got a few genuinely unsettling moments (and there's this one sequence in a more recent episode with a very, very nasty sound that occurs suddenly that actually made me throw my headphones off - yikes!), but mostly it's just weird and off-beat and entirely lovely. It's more surrealistic than it is horrific.