@dlowan, yeah, I'm still in Iraq. Currently at Joint Base Balad, which I believe is the largest military air base outside of the States.
@Robert Gentel, I suspect it's less about insurance and more about scale. There are plenty of American contractors flying into Iraq, but for the most part they aren't taking commercial flights. An exception is the Gryphon Airlines
flight from Kuwait international, which is expensive (~$1200, round-trip) but worth it since Gryphon flies to Sather Air Base (the US military side of Baghdad International) rather than the commercial airport, which means significantly less hassle and more safety (assuming you have a military CAC card or other appropriate paperwork that allows you access to the base).
As others have noted, you can get cheap flights from nearby places (e.g., Jordan or Cairo). I haven't shopped around but wouldn't be surprised to see a round-trip airfare into Baghdad International by a Middle Eastern carrier for less than $500, although it might be difficult to get the best prices unless you buy them locally.
With the right paperwork, there are also free US embassy flights from Jordan to the IZ (downtown Baghdad), I believe (possibly among other options), and military flights from Germany and Ali Al Salem Air Base, Kuwait (which you can take a taxi to from Kuwait International) to Sather Air Base.
Some people here have mentioned flights to Erbil. If you were interested in visiting Iraq as a tourist, that would probably be the place to go. It's within Iraqi Kurdistan, and as a result it would be far safer than most of the country for a Western tourist. Military contractors in Erbil live within the city itself, rather than staying within military bases as is the case elsewhere.