There are wordpress plug-ins that do a similar thing, I looked at these.
(Sermon Browser does most of what I wanted).
Problem is that each 30-40 minute talk uses 25 Mbytes of storage as well as the bandwidth used by people playing the talks.
It means that there can be a hosting cost / management impact.
Uploading files that size usually means FTP, followed by adding a record that points to the uploaded file.
(I have used a dropbox public shareed folder to host the MP3 files, which worked reasonably well).
The sermon.net option is just simple to use, a one off paste the code into your site.
Adding a talk is achieved using a form on thier website to upload the mp3 and associated powerpoints / pdfs.
(I have given this part to our AV team, takes 5 minutes following a meeting).
Only downside is being flash, the player does not work with Apple devices,
They need a Podcast playing APP or subscribe to the RSS feed in Itunes.
My wishlist would be for a HTML5 podcast player plug-in for WordPress which would fix this neatly.