It's not specific to the periods you mentioned, but Mike Duncan's old The History of Rome and current podcast Revolutions are both engaging, accurate and generally pretty well executed. He's a bit less bombastic than Dan Carlin (who I also very much enjoy), but personally I find his style very engaging and entertaining. He also makes a point of making on-air corrections when someone points out a detail that was presented incorrectly.

The early episodes of The History of Rome are a bit rough both from a stylistic and technical standpoint, and also much shorter, but by about episode 15 most of those issues are smoothed out.

BBC Radio 4 produces a range of history podcasts worth listening to. I recommend Making History and In Our Time. They also make a biography series called Great Lives.

Ottoman History Podcasts are very informative. They, like the AskHistorians podcasts, are conversations between a host and a guest, who is always an academic of some kind.

The History Of WWII Podcast by Ray Harris Jr. is quite good for that era. It takes a lot of cues from the incomparable History of Rome podcast by Mike Duncan, and does a great job of giving you good, meaty info without being dry.

The BBC Radio series "In Our Time" is also available as free podcasts, and they often cover historical topics. Each episode features a panel of 3 experts, who often disagree which makes for very interesting discussion.

Both of those are available for free on the iTunes podcast store as well.

New Books in History and the associated podcasts on the New Books Network (Russia and Eurasian studies, American studies, etc.) is a good way to keep up on current scholarship in the historical profession. The audio quality is not terribly great (it's pretty much a recording from Skype) but they are good way to get a digest of an entire book in about an hour.