Where does Amazon Music store files?
I ultimately found the m4a files in /mnt/extSdCard/Android/data/com.amazon.mp3/files/Music

The trick, however, is that they are hidden files.  They do not appear when viewing the folder with either the native file manager or RootExplorer (even when "Show Hidden Files" is checked).  They are found only by using RootExplorer to search the device for "m4a" files.

At least now I know they are on the SD card anyway.