

Chess is not a natural fit for podcasts: with no visuals available, any detailed discussion of positions would place such a high cognitive demand on listeners that a large part of its potential audience would be unable to follow the conversation. So, what can you talk about with chess players if you can't talk about chess positions?
Actually, quite a lot.
Podcasts are an excellent medium for interviews. If a chess player has something interesting to say about anything other than a specific chess position, and if the interviewer can get those thoughts out, then a chess podcast can be as good as any podcast.
Perpetual Chess Podcast
The only all-chess podcast on this list, and possibly the only one on the internet. Host Ben Johnson (FIDE 2174) lives and teaches chess in Pittsburgh. He has produced 23 podcasts to date (May 13, 2017). Most of his guests have been chess players in the US, but he has had a few from around the world, including GMs Peter Svidler, Jon Ludvig Hammer, Simon Williams, Jan Gustafsson, and IM Christof Sielecki.
Johnson isn't a gifted interviewer, and so most of his conversations follow the predictable pattern of generic questions: when did you learn, who were your coaches, what is your biggest success, favourite books, what are your upcoming plans, etc. This is solid start to any interview, but what he needs to make his podcast stand out is to go off-script with good follow-up questions. For instance, when Peter Svidler mentions that as a youngster he got 4 hours of chess lessons a day, I want a follow-up question asking for details about those lessons; e.g. was it four hours in a row (which is VERY long for any preteen) or were there breaks? When current US Women's Champion Sabina Foisor says that most of her opening preparation for the 2017 US Women's Championship didn't get used, but that it didn't matter... I want to ask her whether this is a reason for her to conclude that opening preparation is over-rated and whether she'll spend less time on opening prep for her future events. Instead, we get the usual moaning about how nobody likes "the fact" that all the top competitors "have to" spend so much time preparing openings. These are opportunities to take personal experience and grind it against the cliches and (maybe) find what's really true; it's a shame to see them slide by.
So these podcasts stand or fall based almost entirely on the quality of the interviewees. Some of them, like Svidler and Yermolinsky and chess.com organizers Danny Rensch and Greg Shahade are seasoned raconteurs, and could engagingly monologue if necessary. Others need more drawing out, and don't always get it.
As a test run, I suggest the following:
https://www.perpetualchesspod.com/
Josh Waitzkin, interviewed by Tim Ferriss
Tim Ferriss, author of "The 4-Hour Work Week", has one of the world's most popular podcasts, named iTunes Best of 2016. His podcasts are distinguished by their length -- typically, 90 minutes to 2 hours+ per episode -- and their willingness to go way into the obscure details of his subjects' expertise.
Ferriss is friends with Josh Waitzkin, the chess prodigy and inspiration for the book and film "Searching for Bobby Fischer". Waitzkin is having an interesting life: he quit chess and became a world champion at Tai Chi push hands, and is professionally a personal advisor to investors and sports stars, and the author of the book "The Art of Learning".
He has appeared on the Tim Ferriss podcast twice:
Episode 2: Josh Waitzkin
http://tim.blog/2014/04/22/tim-ferriss-podcast/#more-12146
Josh Waitzkin, the Prodigy Returns
http://tim.blog/2016/03/23/josh-waitzkin-the-prodigy-returns/
Bonus:
Tim Ferriss podcast with Adam Robinson, chess master and co-founder of The Princeton Review (for preparing for SAT exams). Among other things, Robinson discusses studying chess with Bobby Fischer in 1972.
http://tim.blog/2017/02/02/lessons-from-warren-buffett-bobby-fischer-and-other-outliers/
Magnus Carlsen, interviewed at the 2017 Milken Institute Global Conference
Magnus Carlsen in interviewed and then gives a 10-minute clocked simul against some of the big-brained and big-walleted attendees at a session called "Chess and the Art of War: Strategies That Win" at the Milken Institute's Global Conference in Los Angeles, May 1, 2017.
While technically not a podcast -- it's a video of Magnus doing an interview then playing a clock simul -- I converted the video to MP3 and listened to it as a podcast, and nothing of value was lost in the process. As usual, when a top player is interviewed for an audience of non-chess-players, the interviewer tries to extract some bits of wisdom that might transfer from chess to other domains. The one that should, but probably won't, is how often Magnus says "I don't know" when asked a question outside his brilliant but limited area of competence. What will more likely be remembered is his attitude to winning and losing: "I've always been told that you should win and lose with a smile, but since I became a professional, I've thought the best way of learning to deal with defeat is not to lose."
https://www.youtube.com/watch?v=u77Ld9UZqYM
http://www.milkeninstitute.org/events/conferences/global-conference/2017/panel-detail/6933
Garry Kasparov, interviewed by Tyler Cowen
Tyler Cowen is an economist at George Mason University. He asks plenty of questions about chess -- including some detailed ones about Kasparov's brilliant 1999 sacrificial win over Topalov -- but the focus is mainly on Kasparov's new book "Deep Thinking", which uses his loss to Deep Blue as a personal entry point to a discussion about artificial intelligence, and Cowen asks some wonderfully provocative questions.
https://soundcloud.com/conversationswithtyler
Garry Kasparov, interviewed by James Altucher
James Altucher is an NYC-based hedge fund manager, venture capitalist, and author of 11 books about investment and personal development, including the wonderfully-titled "The Seven Habits of Highly Effective Mediocre People", which includes the following line in the introduction: "I've written some books, most of which I no longer like."
The James Altucher Show podcast features interviews with a range of high performers. His interviewing style may be an acquired taste -- imagine if Woody Allen was curious about the modern world -- but he has a high rating on podbay.fm, so there are enough people who have that taste.
His latest podcast is with Garry Kasparov, who is promoting his new book "Deep Thinking". Altucher has a surprising personal link to that Man vs Machine match: he was the chess master who beta-tested the ChipTest program for Feng-hsiung Hsu, before IBM bought it and the rest of the development team and renamed it Deep Thought, the immediate predecessor to Deep Blue. Altucher is clearly interviewing one of his heroes, so this podcast verges on fan-boy territory, but it's also a rare case of Kasparov being interviewed these days by a knowledgeable chess player, so it keeps a very long way from the usual "what's-your-favourite-piece?" garbage.
In fact, after the podcast Kasparov played a game with Altucher, sportingly reaching down to a vinyl board on the floor, a game you can replay below.
http://www.jamesaltucher.com/2017/05/garry-kasparov/
New Kasparov Game: Altucher v Kasparov
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