In-season, I recommend the Grantland NFL Podcast. IMO, it's by far the best football podcast out there. And FWIW, they spoke briefly about Deflategate when it broke, agreed that it was a ridiculous non-story, then agreed not to speak of it again. Between the Super Bowl and TC, though, they pretty much shut down shop, only emerging for some podcasts around the start of FA and the draft. So this is pretty much their offseason.
For Patriots-specific info, PFW in Progress is pretty decent. I don't love it, but at a bare minimum it's better than the crap that WEEI and 98.5 put out on the airwaves. The Around the NFL podcast is also okay sometimes, though I'm not personally a fan. There's also the Dave Dameshek Football Podcast, which I do like. He can kind of grate on you after a while, so I don't catch all of his podcasts, but when I'm in the right mood I do enjoy them. Ross Tucker also has a podcast that can be pretty good, considering that he's a former player and pretty articulate guy.
Once Bill Simmons resurfaces, his podcast (dunno if it will still be called the BS Report) is generally really good. Granted, his focus is divided among a bunch of stuff (his and Zach Lowe's NBA podcasts are in a lot of ways what got me back into the NBA), but during the NFL season he has a lot of football content. Given that he's gone from ESPN now, I don't know how that will affect his podcasts going forward. If it means no more Cousin Sal, that would be a bummer.