This is a pretty extensive resource on the why's and how's of using audio in the writing classroom. There are lots of good ideas and rationale here in this Kairos piece.
This handy podcasting and audio sharing site is both web-based and in the iTunes App store (free). You can easily upload audio files that you have already made or record directly into the site from the website or the app. Sharing out is easy -- with a link or embed code. The account is free, but there is a limit on number of minutes before they start pressuring you to upgrade.
This may be one of the simplest podcasting tools out there right now. You don't need to register. You just need a microphone. Click record, start recording, and then share out the link, embed code, or download the file. The downside is that the quality of the podcast is minimal.