I checked Microsoft's recently launched file storage service and I was surprised to see there's no way to view your files online. Not even to listen to your MP3s. And Microsoft's service is not a singular case: there are many sites that link to an MP3 and expect you to download the file and open it in your favorite player.
But there are also Flash players that let you listen to any MP3 file available online: one of these players was created by Google to use it for Gmail's MP3 attachments. A Greasemonkey script (you need Firefox + Greasemonkey to install it) can add the player next to each link to an MP3 file. Or at least to a file that has the .mp3 extension.
You could also add this code to your site to include the player (replace [MP3_file] with the proper URL of the MP3 file):
Now if only we could have a player for other file formats like Ogg Vorbis or Windows Media Audio...