Hello, [In Our Time] (http://www.bbc.co.uk/programmes/b006qykl) hosted by Melvyn Bragg. Three leading experts are asked on to discuss a particular theory, idea, time or invention in History over forty minutes. Bragg can be a bit cantankerous, but that's ok because he needs to reign in some eggheads who just love the sound of their own voice.
I cannot speak to exactly how historically accurate it is but I am currently listening to Revolutions and it is really engaging. The guy who runs it does not constantly provide sources but he does quote landmark historians like S.R. Gardiner. He provides a bibliography for each revolution.
Is this answer appropriate for our subreddit?
Mike Duncan who does Revolutions as mentioned by u/6767w also did The History of Rome. I also liked Lars Brownworth's Norman Centuries and 12 Byzantine Rulers.
a ton. check out /r/historypodcast, /r/historypodcasting for lists of history podcasts and search /r/podcasts for a lot of people asking variations on this question.
I'm a big fan of podcasts that are open about what quality sources they are using so i'm a big fan of when diplomacy fails (especially when it stops being episodic and gets into long series on things like the 30 years war or World War one) and History of the Crusades (especially after first few episodes). History of Byzantium has gotten a shout-out or two on a similar question from a byzintine flair here and has a few interviews with modern historians, history of philosophy is high quality and run by a philosophy professor, edit: /u/shlin28 is the guy i was thinking about. Similarly backstory is a creation of 3 history professors focused on jumping around for themes/topics across american history (each one of the professors covers "a century" as in they introduce themselves as the 18th, 19th and 20th century guy).
In our time is a radio program/podcast which often takes 2/3 historians and talks to them about a specific topic,
there are a couple of history of english podcasts (BHP, history of England, history of English aka history of English language) which are good and enjoyable especially if you, like me, really lack a basic factual in the history.
if you want to branch out you can also try some itunes U podcasts/recorded lectures. I'd argue at this point those are a bit undervalued.
if we're going for what seem to be less well known podcasts I'd point out history of bulgaria podcast. its interesting to get a bulgarian centric rundown of this period after getting the more "normal" telling that places the byzantine empire at the center.
The History of the World in 100 Objects from the BBC was a great listen.
If you are a fan of European history, then I would very much recommend, "The history of Denmark" podcast. It's a pretty new series that covers the entire Danish history from the oldest kings, to the newest. He has just begun covering the reign of Christian IV. The quality of the podcast is very high, and I think the low listenercount is almost a crime.
If I had to say one criticism, then it would probably be the sound quality and the low volume of his voice. But otherwise, a great podcast series that deserves more listeners
old questions on the same topic:
https://www.reddit.com/r/AskHistorians/comments/33rpmn/i_want_to_learn_by_listening_what_history/
https://www.reddit.com/r/AskHistorians/comments/17dws7/what_is_your_favourite_history_podcasts/
https://www.reddit.com/r/AskHistorians/comments/2puhk2/are_there_any_good_critical_history_podcast/
https://www.reddit.com/r/AskHistorians/comments/3dywnp/what_history_podcasts_do_you_listen_to/
https://www.reddit.com/r/AskHistorians/comments/2iiyaz/what_good_history_podcasts_do_you_recommend/
I am an avid listener to history podcasts and I am currently enjoying the following in addition to the one's already mentioned. The history of England by David Crowther The History of China; The History of Japan (formatting edit)
History of Rome is pretty good. It's an extremely long running podcast covering Rome from the founding to it's final Emperor's exile. Episodes are approx 8 min each.
Edit: gave a better answer and a link.
History of Britain is phenomenal and probably the single best source of information available on Britain's dark ages and period of Danelaw.