None of the household-name GCC carriers fly to Bagdhad. I can tell you that I wouldn't be keen on doing so either.
However, out of Dubai's Terminal 2 (the non-glitzy one without the fancy palm trees and Duty Free shops) you can get to all of the "dark back-alley" destinations in the region using airlines you never heard of and on aircraft that you though were too fuel-inefficient for anybody to operate anymore. Kabul, Bagdhad, Kish Island and Mogadishu becon.