No, not in Fallujah. Fallujah is a mess of Saddamites (these kids in the mob chanted stuff about revenge for Saddam. Tough... he's still in crowbar motel), Islamists, and smugglers who were getting fat on the old regime. All three of these groups hate us for various reasons. But the Islamists here (and the people in general) are of the Sunni branch of Islam, which is the largest branch worldwide, but a minority in Iraq.
The Iranian-sponsored mullahs are Shiites (and the Shiite term for an authoritative mullah is ayatollah. The one who is in the news the most, and the guy that gives us the most lip, is a young buck named Muqtada al-Sadr, whose father was a Shiite ayatollah martyred by Saddam. For all Sadr's publicity-seeking stunts, he has fewer followers than an older imam, Ayatollah al-Sistani. The Shiites are strongest in the south and east (Basra, Nasiriya) but Sadr's strength is in a Shiite slum in Baghdad.
The Shia branch is strong in Iraq and Iran but in decline in most other places and in Islam overall.
The Sunni islamists hate the Shiites even more that they hate us: we're infidels, but the Shiites are worse, heretics. For the historical background on the split between Shia and Sunni Islam, check any net site on Islam.
There are at least five different major Shiite sects, and Lord knows how many Sunni ones. Both branches of Islam encompass sects of every degree from casual, unobservant believers (think Episcopals) to crusading fundamentalists (think Pentacostals, or Puritans in Cromwell's day might be more apt).
They are quite hard to understand, but we have all seen, and borne, the consequences of not understanding them.
d.o.l.
Criminal Number 18F