Normally, I prefer to do the process myself: that is, I extract the audio with a program (VLC) and then use Audacity to convert the file to whatever audio format I like. The advantage of this approach is that I stand on my own feet and can get my .mp3 file all the time and the way I like it. The downside is that, depending on the distro I am using, I might not have the tools available.
If you do not know how to use those tools (or ignore how to use the Linux terminal to achieve that purpose easily), you can always take a point-and-click approach to convert .flv videos to .mp3 files thanks to a Firefox add-on called BestVideoDownloader 2 (you can get it here). This add-on relies on a transcoding Web service called converttomp3.net.
Once you have installed BestVideoDownloader 2, go to your desired YouTube video, play it and scroll down to look under the title. Right next to the like/dislike buttons, you will see the downloading icon of BestVideoDownloader 2.
Click on it and a drop-down menu with format options will appear. Select the one you prefer. I used MP3 192 Kbps:
You will be taken to converttomp3.net service, which will indicate that the conversion of the file is in progress:
The conversion will finish and the service will ask you to specify the .mp3 file tags. Simply click on "CONTINUE":
This will take you (finally!) to the download page. Notice that the service will only store your .mp3 for two hours.
This process is certainly convenient, but it has a main drawback: you depend on the service to be up and running for your conversion to work. Another potential problem can be that, depending on the version of Firefox that you use (or if you update the browser in the future), the add-on might stop working until it catches up with the browser version.