I used to hate anime music, it was so cheesy. Though I love Urusei Yatsura and Ranma 1/2, the music is abominable. Boy has it gotten better since the 90's.

Tank! by Yoko Kano and The Seatbelts from Cowboy Bebop

Rise by Yoko Kano from Ghost in the Shell Stand Alone Complex

Ride on Shooting Star by The Pillows from FLCL

Hare Hare Yukai by Aya Hirano from The Melancholy of Haruhi Suzumiya