Don't mop yet.
That's the most useful thing we can tell anyone standing in a freshly remodeled kitchen looking at a dusty tile floor. Drywall dust, tile-saw slurry, and fine sanding dust are mostly gypsum and mineral powder. Add water and they become a thin paste. That paste flows straight into the grout lines, dries there, and turns grout that was fine before the project into grout that looks a shade darker after it.
The order that works
Wait until the dusty work is actually finished. If someone is still sanding drywall in the next room, anything you clean today is getting a fresh coat tomorrow. Let the dust settle overnight with the HVAC off or the vents covered, if you can.
Clean from the top down. Wipe the cabinet tops, window sills, light fixtures, and door frames, and change the HVAC filter. Every bit of dust up there will end up on the floor eventually.
Then get the dry dust off the floor while it's still dry. A vacuum with a good filter (a HEPA shop vac is ideal) is better than a broom here; sweeping throws the finest dust back into the air, and it lands right where you just were. Run the crevice tool down the grout lines. Go over the floor twice.
Only now does water come in. Work in small sections with a pH-neutral cleaner and a damp, not dripping, microfiber mop. Rinse the mop head constantly. A bucket of cloudy water is just spreading dust around in a thinner layer. Change it often.
Let the floor dry, then look at it in daylight, low and from the side. Wet tile hides a lot.
Cloudy tile after cleaning
If the tile still looks hazy after two careful passes, it may not be dust at all.
New tile installations often leave a thin film of grout residue on the tile face, called grout haze. It's cement, so it doesn't mop off the way dust does, and scrubbing harder just scratches the glaze. Check with whoever installed the tile; they usually have a specific haze remover and know how long the new grout needs to cure before anything acidic touches it. Fresh grout is soft for a while, and aggressive cleaning in the first few weeks can damage it.
One more on new grout: if you're unsure the grout has cured, wait. A week of living with a little dust costs nothing. Grout that gets damaged while soft has to be dug out and redone.
Older homes are a different caution. If the remodel disturbed old flooring, paint, or ceilings, ask your contractor what was in them before you start sweeping. Some older materials need professional handling.
What else caught the dust
Remodeling dust travels well past the room with the work. It drifts down hallways and settles into carpet, area rugs, and the couch, and pets carry it around on their feet and coats. A carpet that looked fine before the project can feel gritty underfoot a week later.
Fine mineral dust in carpet is abrasive. Vacuum it slowly and repeatedly with a good filter, and if it still feels gritty, a professional carpet cleaning will pull out what's settled deep in the carpet. Rugs and furniture that sat near the work zone usually benefit from the same visit.
Quick answers
Using vinegar on new grout
We wouldn't. It's acidic, and new grout is still curing. Vinegar is also hard on natural stone.
Grout that looks darker after the remodel
Usually slurry or dust paste settled into the lines during cleanup. A deeper clean typically brings it back. If the grout itself was installed in a different color, cleaning won't change that.
When to call us
When the tile is clean but the grout lines still look dingy, or when you've mopped three times and the floor keeps clouding up. We go after grout with a machine buffer, which drives the cleaning solution down into each joint far more evenly than a hand brush can. The loosened residue then gets pulled off the floor and the tile is rinsed. Before any of that, you'll get a walk-through of the room and a price. There's more on how it works on the tile service page.
Finishing a project is expensive enough. The $45 tile coupon on our coupons page helps a little, and the scheduler works whenever your contractor hands back the keys.