Paint rollers are gentle on popcorn ceilings, but you can choose a thick-nap one for maximum protection. Wrap duct tape tightly around it, sticky side up. Unlike brooms, this combination won't damage the ceiling. Gently move the roller over the ceiling. The dust will stick to the tape. When it stops working, try replacing the tape. The flexible nature of the roller and tape should also help you get into corners. If you don't mind causing flakes of the ceiling to fall, a broom can work. Make sure your standard household broom is clean so it doesn't reintroduce dirt. As gently as possible, brush it across the ceiling. Flakes will come off, but so will the dust spots. Wear a mask and eye protection when you do this. Popcorn ceilings from the 1970s and earlier may include asbestos, which is harmless when left alone.

Summary:
Wrap duct tape around a paint roller. Roll over the ceiling. Use a broom.