Last year Google announced its podcasting service called Google Podcasts. Android users have had the service, but not iOS users. Today iOS users can find and listen to podcasts within Google search (via Engadget).
Google Podcasts is available on the desktop as well. Just go to www.google.com, search for a podcast, and scroll down until you see play buttons. It will take you to podcasts.google.com to play them. However, going directly to podcasts.google.com won’t work.
If you sign into Google, it will remember your last position in a podcast and sync it across devices, so you can start listening where you left off.