These are the best WordPress podcasts I've found. They have content for WordPress site owners, designers, and developers. My favorites are those that are focused on education rather than news. I listed them in order of preference.
If you like these, you should read my other posts about podcasts. I’m always looking for more, so please add your favorites in the comments!
WPwatercooler has become my favorite WordPress podcast. It's great for the hosts' range of opinions and advice. It features roundtable discussions about providing client services, WordPress development, digital marketing, and the WordPress community. Sometimes it's hard to understand what's said because people talk over each other. I love that they limit each episode to 30 minutes. iTunes
Your Website Engineer includes fairly detailed instructions for designing, developing, and maintaining WordPress sites. It used to be very focused on WordPress, but recently it's covered a range of digital marketing topics. I hung out with host Dustin Hartzler at WordCamp Grand Rapids 2012 and 2013, and I've had the honor of being interviewed in Episode 104: Become a Full-Time Freelance Developer. iTunes
The Matt Report is a series of interviews with people who run businesses around WordPress, usually providing client services or selling products (themes, plugins, maintenance, etc.). The guests often share valuable business insights. iTunes
WP Elevation focuses on the business side of WordPress, featuring interviews with WordPress entrepreneurs and businesspeople. It's more about business than WordPress. iTunes
Kitchen Sink WordPress episodes are refreshingly short (often 10-15 minutes), with tips, tricks and tools, and interviews with WordPress users. I met Adam Silver online a few years ago, through Dustin Hartzler, host of Your Website Engineer. iTunes
The BlogAid Podcast curates tips on WordPress, content marketing, SEO, and digital marketing. Most of the tips are mentions of posts or other content online, so you need to look them up after listening to the podcast. It's nice and short (10-12 minutes). Almost every podcast includes Google+ tips, and I wish they didn't. iTunes
WP Unicorn Project includes guests who discuss WordPress design, development, content, and business. iTunes
WP Bacon includes interviews with WordPress experts about technical and business topics, as well as discussions about the WordPress community. I didn't like WP Bacon when it first came out, but I recently resubscribed because I'm more interested in the topics they're covering now. iTunes
Please add your favorites in the comments!