How to properly clean a rough concrete floor?
Mopping is great. It shouldn't be that dirty subsequent times, so it won't take as long from here out. Sweeping or vacuuming before mopping will help reduce the chance of needing more than one pass.
If you want a less manual mopping, you can get a mopping floor machine, steam floor machine, or a robotic mopper (like a roomba). Or you can get a big yellow rubbermade janitor-level mop and bucket for about $75 that will make mopping a lot quicker.
Adding a floor treatment like epoxy would also help reduce the permeability and increase the smoothness of the floor, which makes sweeping and mopping more effective. You can also just get cheap indoor/outdoor carpet for the flattest parts to cut down on what need scrubbed and mopped.