Despite what some websites may tell you, Yuzo Koshiro did not do the music for Shinobi III. Instead, three people are listed in the credits for it, these being Masayuki Nagao, Hirofumi Murasaki and Morihiko Akiyama. I think they did a great job with this soundtrack, and like the music even more than any of the other Shinobi games.

Track title:
Used in:
Game Over
Game over screen
Getufu
Continue screen
Ground Zero
Credits sequence
Hassou!
Sub-boss battles, Round 6 boss battle
He Runs
Round 1-2
Idaten
Round 2-1, Round 5-1
Inner Darkside
Round 3-1
Izayoi
Round 6-2
Japonesque
Round 1-1, Round 6-1
Mandara
Rounds 1 and 6 boss battles
My Dear D
Round 3 boss battle
Ninja Soul
Round 2-1
Round Clear
Round clear screen
Rush and Beat
Round 5-2
Sabi
Life lost
Sakura
Ending sequence
Shadow Master
Final boss battle
Shadows
Rounds 2,4 and 5 bosses
Shinobi
Intro sequence
Shinobi Walk
Round 4-2
Solitary
Round 7-1
Stage Clear
Sub-boss defeated
Storm Wind
Round start screen
Trap Boogie
Round 3-2, Round 7-2
Wabi
Final life lost
Whirlwind
Round 4-1