This is not so much about podcasting, but it could be. Just let me explain.

Blogger is now allowing private blogs. The folks reading and participating need Google accounts, but they are free and no one ever need use the other aspects of the account. This is a great improvement on the service for educators, because it means classroom blogs can be set up and limited to only pre-approved users.

This can allow a similar limit for podcasts. You can create audio files, post them to a web server, and place links in your private blog. Now, unless someone is particularly tenacious, the traffic will be limited to only the folks you want to hear.

I still like WordPress, and that is another option, but the prevalence of Google means that many of your students may already have Google accounts, eliminating some of the initial startup issues.