Sponsor the local baseball team: http://www.mcclatchydc.com/homepage/story/72174.html
Sports, especially team sports, have an ability to bring people together which is second only to music. And every time an Iraqi watchs their baseball team play a game now, they're going to know that some nice people in the USA donated the equipment. For the price of some sports equipment, you just won more hearts and minds than you have in the entire occupation so far.
So, me being me, let's take it over the top and sponsor the entire Iraqi sports community. I'm sure Iraqis play football (soccer to you on the wrong side of the Atlantic), the whole world plays football so let's send them some of Dunlop's best balls, a bunch of studs (cleats) in assorted sizes and some shorts and shirts. I'm pretty sure Iraq must have a basketball scene so lets send them some hoops and balls, a few pairs of Nike's best. In fact, let's help Iraq organise a Pan-Islamic Games. I'm 99.9% sure that the cost of sending a few plane loads of assorted sports equipment is going to be a fuckload cheaper than keeping troops there until the end of time. A lot of Arabic countries have a great deal of national pride. You're not going to change that, you're never going to grind it down so let's channel it into something positive.