At first the Amazon prime music app refused to install on my rooted Fire Phone.
I just installed the APK and it worked fine.
Then was just playing around with this today. Found the downloaded MP3 files are in a hidden directory. com.amazon.mp3 was hidden. (nice try Amazon).
It was a matter of clicking "show hidden files" in linux and duh; there it was.
data/com.amazon.mp3/files/music etc, etc