In my experience, it's simple, reasonable in the amount of time it takes to do, but not quick in time on the clock, due to waiting.
I take multiple passes at the wall with dry cotton towels (think terry towels from Costco)
First take one dry cloth and wipe a whole wall. I avoid moisture, since with any significant water, the mud will move around and you will lose the smoothness you generated by sanding.
This first pass is more of a mass removal than anything. So there will likely be a lot of dust that falls to the floor. I wait for that to settle - vacuum up after a bit (30 min is fine). Let the air settle overnight. All of that assumes you have a HEPA air purifier running to take the dust out of the air - otherwise a lot will settle back on the walls.
The second pass I'm more careful about changing over to different cloths - take a pass at a small area (12-16 sq feet or so), then flip the cloth over and do another area. Refold it so you use all sections of the towel then switch to a new towel.
You're brushing the wall (you can be gentle). If you scrub, you can leave streaks in the dried mud/joint compound.
After two passes my experience is that the walls are as clean as they will need to be. You could still touch the joint compound and get some "dust", but that's likely because your finger is abrasive and you're removing a layer, not because it's actually dust on top.
Note it's a lot better for your washing machine if you rinse the cloths in a bucket and dump the water out in your back yard rather than throwing fully saturated with drywall dust items into your washing machine.