This is for educational purposes only. It is your responsibility to verify whether or not YouTube allows you to do this.
I've recently found a playlist I wanted to "git pull" from YouTube, but realize it would take too long to use tools such as Keepvid, or any other MP3 fetchers online. Then I recalled that there is this neat little tool called youtube-dl, and decided to learn how to use it in order to download MP3s from YouTube.
To my big surprise, it also allows us to download entire playlists.
As a result, I've spent some time to write down a short step by step guide for Windows, that should help us download entire youtube playlists, as well as individual MP3s from youtube videos.
I expect to append a similar guide for Linux in the same gist file, once I get the time.