You may be seeing a prompt in the lower left corner when clicking on an audio file that says, "Downloading file".  If this is the case there are some simple settings that need to be changed.

Windows Media Player users- Open Windows Media Player and click the small drop down arrow in the upper right corner of the screen. Click "tools", and then "options", and then the tab marked "file types".  Check the box next to "Mp3 audio file(.mp3)" and enjoy listening.

Quick Time Users-Open QuickTime and click "edit", then "preferences", and then "QuickTime preferences".  Click the tab marked "file types" and make sure that the "Mp3" and "Audio" boxs are checked and enjoy listening.