I went to Iraq last year and did some limited caching. My main base was very small however, and didn't have any caches. From what I see there's only 280 caches in all of Iraq, so you could just do a PQ for all of Iraq. Or if you don't want that many, just find one on your base and run a search for all caches within 20-30 miles of that. I don't think there's many caches that are not on base so you don't have to worry much about that, but even if there were, don't think you'll be doing any caching off base in Iraq.
I just used the same stealthy techniques I use anywhere else. Carry a clipboard to look official or go when there aren't any people around, you won't get many questions. If you're on one of the larger bases like BIAP or Talil, there are events sometimes as well.
Don't think there's any at Udari, but there are some at the other bases in Kuwait that you can easily pick up.
Keep your head down, good luck and happy caching.