As a companion to Shane's post—since I really liked the idea—these are my top five individual songs from Japanese bands. I first heard most of them while watching anime, except for the L'arc~en~Ciel song which I heard in an HMV in Shibuya when the record came out.
The only rules: no composers, and no long descriptions.
I'm sure Cole will agree with a lot of these, and I hope the Fairy Fore song brings back memories of watching Final Fantasy Unlimited at his old apartment, not entirely sure why we were doing so (perhaps just to hear what "Soil Charge Triad" would be used next, or what would become of the purse-eating, dubiously clairvoyant Fungo).
5. Zetsoubou Billy - Maximum the Hormone
Great chorus. Just ignore the screaming/pseudo-rapping. Love the bridge at 3:08 and the female vocals.
4. My Heart Draws a Dream - L'Arc~en~Ciel
Beautiful piano towards the end. Ethereal.
3. Vivid - Fairy Fore
If sugar was music. Picture dancing chocobos.
2. Bran-New Lovesong - The Pillows
Hard to pick a single song from the FLCL soundtrack. This one stands out for me. It's like watching an orange sunset from a bridge in a small town.
1. A Town in Blue - Asian Kung-Fu Generation
Amazing guitar. Closes off my all-time favourite anime Tekkonkinkreet.