Iraq is something of a special case, because the Iraqi troops abandoned enormous quantities of weapons and ammo, and many warehouses and storage rooms of weapons and ammo were later discovered. Most troops see no harm in grabbing a pistol (although many states require these to be registered) or bayonet. The Iraqi army also issued ceremonial swords to many officers, and these were really popular as loot. As with every war, the officers demand that all loot be turned in, and then hold inspections to make sure everyone has complied. But for those adventurous soldiers and marines, this merely adds a bit of challenge to their determined efforts to get souvenirs out of the country. There are many places to hide your loot. American units came into Iraq with lots of vehicles, each of which provides many hiding places for all sorts of goodies. There's lots of hiding places in other equipment, like generators, field kitchens and the like. Support troops are usually eager to help out with the smuggling, for a share of the loot. So while you'll hear a lot about strenuous efforts to prevent souvenirs from leaving the country, remember to check eBay for Iraqi military items in the next few months.
As with every war, American troops are now faced with the serious business of getting souvenirs out of the country without getting caught. Since World War II, when many troops brought home enemy weapons, flags and parts of uniforms (and enemy bodies) , there has always been that urge to bring back something. But the brass have long taken a dim view of this. For one thing, many of the weapons brought back were illegal in the United States. Many American homes still have a German or Japanese machine-gun or mortar in a closet, courtesy of a resourceful grandparent. In the United States, you cannon posses those weapons without a special federal permit. More troublesome than the weapons is the ammunition. Bringing home enemy grenades or small shells (20mm is always a favorite), often brings tragedy later as someone plays around with the ammo and it goes off.