Move your cursor over the link to the file. Check the destination at the bottom of your browser. If the file extension in the link ends with ".MP3," this is a direct link to the file. Right-click and select "Save link as..." or "Save file..." to download the file. If mousing over the link does not preview your destination, try the next step.
Check the type of player that the website uses to host the MP3s. If each file is marked with an arrow inside a small grey circle, the site uses a simple Yahoo! media player. Right-click the title and select "Save link as..." to download the file. Even if you don't see the arrow icon, a direct link on the page generally indicates that you can access the file by right-clicking and saving the file to your hard drive. If the player is more complex, attempt step three.
View the source for the web page. Click "View" and select "View Source" or "View Page Source." The browser will display the source code for the entire page currently open in your Internet browser. Press Ctrl+F (or Cmd+F on a Mac) and type "MP3" into the search field. The browser has now highlighted all of the instances of the phrase "mp3" on the page. You can scroll through each of these by pressing the tab key and searching the page. Once you have located a URL that ends with the .mp3 file extension, copy the entire URL (beginning with http) and paste it into your browser. If you have located an active link to the MP3 file, your browser should prompt you to save the file to your hard drive. Select a destination for the file and click "OK" to save the file.