find_and_move.sh

find /dir/ -iname "*.mp3" -print
# /dir/   - Search directory
# -iname  - File name search pattern (case insensitive)
# -print  - Display name of files on screen


find / -iname "*.mp3" -exec mv {} /new_dir/mp3 \;
# -exec mv {} /mnt/mp3 \; 
# Execute mv command. 
# The string '{}' is replaced by the file name.
# \; ends /bin/mv command.

# To just move .mp3 files and not directories, use:
find / -iname "*.mp3" -type f -exec /bin/mv {} /mnt/mp3 \;

# Find all directories having name mp3 and move:
find / -iname "*.mp3" -type d -exec /bin/mv {} /mnt/mp3 \;

# For performance you may need to consider using xargs command:
find / -iname "*.mp3" -type f | xargs -I '{}' mv {} /mnt/mp3

# To moves all .mp3 files with special characters in its name such as white spaces try:
find / -iname "*.mp3" -type f -print0 | xargs -0 -I '{}' /bin/mv "{}" /mnt/mp3/

# Above commands will not maintain sub-directory structure. Try replacing mv with rsync to use the same directory structure on the target directory with the -R option:
find / -iname "*.mp3" -type f -print0 | xargs -0 -I '{}' /usr/bin/rsync -avR "{}" /mnt/mp3/