Google has indexed thousands of webpages that are simply directories full of mp3 files. These directories are hosted by many different people all across the Internet.
By adding some commands to your Google search, you can restrict the results to show only pages with links to mp3 files on them. Here is the search query we use:
intitle:index.of mp3 -html -htm -php -asp -txt -pls yoursearchhere
Note on spam: Many sites have figured out that folks do searches like this and so include "Index of MP3" in the title even though they aren't an index of mp3 files. Some sites go even further and use Mod Rewrite techniques to host web pages like "spamsite.com/somefile.mp3" that isn't really an MP3 file.
Our site uses the Google AJAX Search API to make doing these searches easier. Just type your search into the box above, and we add all the extra bits. We also try our best to filter out the spammy websites and provide a tool that quickly scans the page for mp3 files and lists the download links. We even include a Flash player for streaming the MP3s from our site.
Note: Google now streams music direct from search results. To learn more about how to use Google.com itself to find songs (and lots of 30-second and 1-minute demos) to stream, visit Google's Discover Music page.