Here is an article from last July on the Iraqi Sunni Arab tribes and their loyalties:
http://www.hnn.us/comments/14860.html I only gave it a cursory overview, but it appears to touch on a number of issues worth knowing in dealing with that region of Iraq. It does indicate that Fallujah is a stronghold of the Dulaym tribe. The 10 of clubs, Latif Nussayif al-Jasim ad-Dulaymi, captured last June, was from this tribe.