Many bands with the name Pillow exist:

1. Italian band Pillow is a solo project of Giardini di Mirò's keyboardist Luca di Mira. He released album Flowing Seasons under 2nd rec.

2. Belgian band Pillow was founded in january 2004 by a Guillaume Vilain but was quickly join up by four guys from Belgium and France. Together, they produce some depth and instrumental short tracks. These compositions full of various structures go straight to the point.
Played with: The Editors, ¡Forward, Russia!, The Isles; Styrofoam, Giardini di Mirò
Links: pillow.fr, myspace.com/pillowfr, cartepostalerecords.be.

3. There is a band based in the North-East of England called Pillow and you can find their info here: Pillow

4. Greek Pillow were an english-speaking, "Sarah-Records-esque" indie-pop band in Athens, Greece, in the mid-90s.