These are the commands to search and move data on linux
-iname – File name search pattern (case insensitive)
Move .mp3 files and not directories, use:
Find all directories having name mp3 and move:
Above commands will not maintain sub-directory structure. Try replacing mv with rsync to use the same directory structure on the target directory.
You need to pass the –remove-source-files option to the rsync command to delete source file.