Normally what you would see when you go to a listing for podcasts that are in iTunes would be this:
Just have your students install the iTunes Audio Prevgiew/Podcast Downloader extension from the Chrome web store
Now, when you go to your chosen podcast page, the extension will add a Download link next to each audio file that the students can click on and listen to or download to their chromebook or Google Drive. This will also work on your laptop when using the Chrome Browser.
Once on the new tab, you can get the url for that specific podcast to share with your students... Check out the iTunes podcast listing page, for thousands of free podcasts organized into dozens of searchable categories. Once you find a podcast you'd like your students to listen to, you can share the URL with them by posting it on your website, in your LMS (Moodle, Edmodo, Schoology), or just email it to them. Unfortunately this extension only works for audio podcasts and not video podcasts.
Update - 10-10-14
Aside from iTunes, there are many Chrome apps for listening to Podcasts. There are too many to mention them all, but just based on ratings in the Chrome web store, here are 3 you may want to consider
Cloud Caster is a podcast player for Chrome that keeps track of all your podcasts in the cloud.Cloud Caster can be used on your Chromebook, PC and Mac as well as on almost all mobile devices running iOS, Android, Windows Phone or Blackberry. This means you will always have access to your podcasts and have a playlist in sync. Cloud Caster remembers where you left off on one device and lets you resume from that position if you continue to listen on another device. You can start listening to a podcast at home on your desktop computer, keep listening on the bus with your mobile device and finish the podcast on your computer at work/school when you get there.
Airing Pods as a Chrome web App is a shortcut to launch airingpods.com.
Airing Pods give you various ways of finding Podcasts (Shows). Comes with an easy to use Podcast Player, which plays your podcasts all in the browser.
No plugin/software to install, and it make this a well suited podcast manager for your Chromebook as well.
ShortOrange is a cloud-based cross-platform podcast player with which you can listen to your favorite podcasts via the ShortOrange Chrome app, online via our web application, or with our iOS app (android coming soon!). In addition, ShortOrange syncs your podcast data between all platforms via the cloud to enable a seamless transition between devices. ShortOrange is the ultimate app to listen, subscribe, and manage all your favorite podcasts in the cloud.