You might try some sand, and sweeping it up. Seems odd that they want it cleaned and that using a pressure washer is not allowed. Someone must have torn up some green concrete on the contractor before.
I ran into this problem, I use a scraper, it's time consuming but I have found that works on really green (soft) concrete. I tried heat, low pressure, chems, etc... Good Luck!
You can actually use this best way to keep hardwood floors clean like using canister vacuum made for use on hardwood floors. Features to keep an eye out for include strong suction on dust and dirt's from your drywall.