To get the ball rolling, here are some of my favorites:
The Bike Shed is a podcast sponsored by thoughtbot, run by thoughtbotters, talking about tech challenges encountered during the work week. I've gained insight into so many technologies from just listening to Chris and Steph chat about topics that showed up during their development week. Conversation often involves Rails and JS, but nothing is off-limits. Elm and Scala have come up a number of times.
A Tasty Treats Podcast for Web Developers.
Hostedy by Wes Bos and Scott Tolinski, Syntax is a podcast for learning about new things in web development. Since Wes and Scott are both educational content creators, episodes vary from topic introductions to somewhat deep dives on more specific concepts. Definitely less "in-the-weeds" than The Bike Shed, which isn't really a good or bad thing.
Conversations with the hackers, leaders, and innovators of the software world
Episodes of The Changelog are generally setup as conversations with interesting people from the tech industry. From developer relations experts to software architects, you may not enjoy the guest of every episode, but there are almost definitely some that will pique your interest.
These are a few tech shows in my podcast player, what's in yours?