OK, it may be a 2 step process depending on the video you are using but here goes:
For this to work well you need to be using Firefox.
Install the "
Download Helper" add-in to the browser (which is free in Firefox)
When you are on a video page you will see the Download Helper icon animate. You can click the drop down to see the available formats. Usually .FLV can be saved with no issue to your system. Once in awhile there are issues with the files, so if audio MP3 is the final goal, MP4 is a better download choice if you have it.
Once you have the file on your desktop (or wherever you want to save it) use
Quick Media Converter. Just select the MP3 conversion option, drag and drop the file in and hit convert. It's that simple.
Lifehacker had a write up on this awhile back, but I don't particularly care for the media converter they recommend. It's adequate but I prefer the one listed above.