How to re-encode iTunes m4a files to mp3
Honestly, for working with audio files quickly, my favourite tool is Fission Audio from Rogue Ameoba. It can open and convert to most popular formats, including the two you're after , and more importantly here I would think, it has a batch conversion mode. It's $32 USD, but it's so polished and works so well it's worth the cost.

But if you really want free I don't think you can beat the command line for converting lots of files, quickly and easily.

To complete these steps you'll need Homebrew installed -- a free package manager for OS X that'll make it easy to get the command line tools you need to do your coversions. See the installation notes for Homebrew on how to install it.

With Homebrew installed you'll need a few command line tools to do the conversion. Namely: ffmpeg and lame. But don't fear! Homebrew makes it very easy to install them. Open a Terminal window and type:

brew install lame ffmpeg


You'll get a bunch of other dependencies installed along with those two things. But those are the two key pieces.

With these tools installed, you can convert an entire directory full of .m4a files to .mp3 like so:

for f in *.m4a; do ffmpeg -i "$f" -acodec libmp3lame -ab 320k "${f%.m4a}.mp3"; done


Just run that from a directory full of .m4a files and watch the magic happen.