Wrap a microfiber cloth around a yard stick...use tape to keep it in place or you will be stuck crawling around on the floor with a bent coat hanger trying to fish it out.
But it works well.
Linda C

Thanks for the suggestion. I have been using a yard stick with a sock held on by a rubber band. Pain in the neck.

LindaC has an innovative, low-cost solution that sure makes sense. If bending is an issue, as it is for me (age, you know) the March 2008 issue of Good Housekeeping shows a new Oxo Good Grips Under Buster that "has a special joint in the handle that bends a full 90 degrees, so the mop can slip easily below furniture that's low to the ground. The included microfiber pad is washable and extra wide, but if you prefer a dry disposable cleaning cloth, just attach one to the head." It's a bit pricey at $25, but my back is telling me it might be worth it. It's supposed to hit the stores this month, but we haven't yet found it in any stores where we live. My guess is that this would meet your clearance requirements, which don't appear to be unusually low.

This may be 9 years too late for you, but now there is the Mini-Mole Robotic Floor Sweeper - which only needs one inch of clearance to clean the floor under furniture.