Highly Recommended
I was in Iraq doing a banking system during the first gulf war (Iran vs Iraq) and the tourism was absolutely fabulous.
Babylon has been excessively done up by the late monster, but it's still pretty impressive -- fragments of brick and tile with cuneiform writing underfoot.
Warka/Uruk is left pretty much undisturbed since it was excavated by Germans before the 1st (world) war -- you approach it across plains ruined by salt, and the first glimpse is the huge piles of potsherd. It was inhabited six thousand years ago, Gilgamesh lived there, and I have never encounted such a sense of intimacy with the deep past. You'd need to check that the bridge had been repaired.
Ur was an army base when I was there, but if you drove past slowly and cautiously, you could peep at the ziggurat where, in theory, Abraham's father worshipped.
It might be wise to stay away from Shia holy sites generally for a while and the Najaf shrines were badly bombed by Sunnis so they won't be what they were, but there's a lot to see if you're interested in Islamic art.
Just south of Babylon you have astonishing alexandrene ruins called Ctesiphon. Out in the western desert there's the huge fortress called Ukkhadir (sp. sorry) which is a very mysterious place indeed and leaves you feeling that you'll never lose the taste of limestone on your lips. On the way back there was a modern resort on the banks of a reservoir -- I wonder if that's still there.
Baghdad itself was rather comprehensively done over by the huns in 14-something so there's little left of the caliphate but the museums (Baghdad and Iraqi national) are something else. I hope that Schliemann's (I think it was him) Babylonian treasures are intact.
I never got a chance to go north, but Nineveh and Mosul are apparently well worth a look.
Just saying the names ought to be enough: Nineveh and Uruk, Babylon and Ur. But if you want a reason for a techy to go, how about pointing out that these were the civilisations which invented the 60-second minute or the 60-minute degree (I didn't say you had to like them...) or (in cuneiform) non-pictographic writing?
OK -- if you want to lounge on a beach and pick up a bit of wossname from catford, it's not ideal, but to plant your feet a little more securely as a citizen of Earth it's probably essential. Beat the crowd.