I don't recommend changing this behavior as leaving the browser to decide which media player to use is the preferred way to handle media download links. If you need an example why you don't want to change this, try downloading the media from an iPhone or Google G1 phone with the following steps below to see how doing something like this inconveniences users.

The most efficient way to do this is to upload the media file to another directory on your web server then create a .htaccess file, change the content-type of the media (e.g. audio/mp3) and set the content type to binary/octet-stream. This will make most browsers display a "File Save As" dialog.

Example .htaccess file:

    AddType  application/octet-stream .mp3

Again I don't recommend doing this, but this is what you can do if you're so inclined to do so.

Your not missing anything, your other hosting company didn't have the content type set for mp3 so users saw a file save as dialog. The behavior you want is non-standard.