More insight into what the Coalition is, and has been, facing in and around Fallujah can be found at the Strategy Page:
What kind of fighting will occur in Fallujah? It will be a game of wits, as well as weapons. The most professional and experienced anti-government gunmen are in Fallujah, and they have developed many countermeasures for the coalition advantages. A major disadvantage is the coalition control of the air, and the constant presence of UAVs, aircraft or helicopters. Too many gunmen have seen their buddies ambushed, or jumped by unexpected coalition troops, to ignore the possibility of a UAV above sending live video of the battlefield to coalition commanders. So the gunmen try to set up movement routes that cannot be seen from the air.
Rugs or sheets are spread across alleys to make this possible, and sometimes even short tunnels are dug. The downside of this is that movement is inhibited. In fact, American troops do not always have a vidcam equipped UAV over the battlefield. But Department of Defense public affairs people like to distribute videos of such operations. The implication that American troops can "see everything" is meant to intimidate the enemy. It does, and slows down enemy movements, often fatally so.
Fighting in Fallujah will be a war of surprise and ambush. Whoever first figures out what the other side is up to will have an edge. The smart money is on the Americans. For while the "insurgents" have received lots of positive press for their unequal struggle, they have by far gotten the worst of it. In thousands of little battles, the anti-government forces are almost always defeated. Most of the time they just flee, but all too often they are killed or captured. Coalition intelligence officers know who they are fighting, and how they fight.
There's more at the link.