I have had success with wet rags. Some areas need to get hit twice before all the fine powder is gone. It is very tedious but it works.

Matt

Lots and lots of clean wet rags as batguys suggested. It's a major pain and it'll take several passes to get all the dust cleaned up.

You say this is new construction. Are you responsible for cleaning during this build?

If so, I would focus my attention on the walls before the painters come in using DAMP rags. Not wet, damp, the walls haven't been primed yet.

I second what has already been said. Vacuum what you can, then use a damp rag to wipe up the rest.