Amumu is highly dependent on teammates, so try laning with your friends for maximum effectiveness.
Cooldown Reduction on Amumu is very strong, but it's often difficult to itemize for it. Grab the Golem buff whenever possible to gain Cooldown Reduction without sacrificing stats.
Despair is very effective against other tanks, so make sure you're in range of opponents with the highest Health.
Avoid bunching up with other allies when Amumu has his ultimate available.
Erratic movement, or hiding behind creep waves can make it difficult for Amumu to instigate a fight with Bandage Toss.
Amumu's Despair makes purchasing primarily Health items a risky proposition.