Chrome:
Verify that your version is current. Make sure you have the most recent RSS plugin (to do this good RSS plugin for chrome and install). Once you have the plugin then you need to select a default podcasts viewer. Chrome will give you a drop down list, Google reader is the default option, it works great but this is more a personal preference (any will work). Make sure you check the option always use this reader to subscribe feeds and you will never have another problem with chrome.
Internet Explorer:
This is a bit more tricky. IF YOU DO NOT HAVE WINDOWS MEDIA PLAYER AS YOUR DEFAULT PLAYER IT WILL NOT WORK. If you have anything other than windows media player selected it will attempt to download a file every time you click on a podcast, regardless of the option you select. If you wish to use Internet Explorer then you must change your default player. Please click this link for instructions, they will vary depending on your operating system so you may have to Google your options.
http://windows.microsoft.com/en-US/windows-xp/help/windows-media-player/11/default-player