1. Open the Firefox web-browser
2. Download and install the Tamper Data extension for Firefox. Restart Firefox.
3. Set up Tamper Data to force Firefox to cache everything, even if the server requests that the data not be cached (which is what Deezer does). So, when you play a song in Deezer’s player with Tamper Data on, the song is forced into the cache - and that’s where you can find it!
4. Navigate to deezer and play a tune.
5. Open the cache folder of your Firefox. You can also type "about:cache" (without quotes) in the address bar and click enter to find the list of Cache Entries and the path to the cache folder.
6. Look for the big entry with the name CacheDiffusion.php; this is the song. Click the link.
7. Refresh the page; if the data size changes, the song hasn’t finished loading, and you should wait.
8. Once the song has loaded, set Firefox to Work Offline, just in case; we don't want to accidentally make any extra requests to Deezer’s servers.
9. Right click on the link and save as songname.mp3. If Tamper Data comes up, okay the request; nothing is being sent out, though, because we’re in Firefox’s offline mode.
10. Cleanup: turn Tamper Data off and turn off the offline mode.

The trick doesn't work for me. When trying to save the link you get the following warning:
Obviously there is some kind of problem with clicking on the link from the cache. Why should one do that? You can wait until the song finishes playing and grab the song from the cache folder of the Firefox.

The problem is that Tamper Data doesn't cache these entries even though the forced caching option is turned on.

This trick is posted here for educational purpose only. You can only record music from services that doesn't prohibit saving music from them! Deezer clearly states that saving music from it is illegal!