Have you ever seen a cleaner dip their mop into what looks like a mud lake? We have, and it's a bit unsettling. But this doesn't mean a mop and bucket is obsolete. After all, a traditional method can still be more "green" by using conservation. You just need to know how to keep your mop bucket clean without using large quantities of water.
The first step in keeping mop water clean is to sweep properly using a clean microfiber dust mop. Microfiber pads can be laundered so you'll always have a fresh pad available. And they do a much better job picking up dirt than a broom or standard dust mop.
Next, never mop without sweeping. Some people try to save time by mopping and sweeping in the same step, but all this will do is contaminate your mop water as well as leave dirt, hairs, and other debris behind.
A common misconception is that a mop bucket needs to be completely full. Why fill a 36qt bucket to mop a couple rooms? You can properly dilute a 1/3 filled bucket just as easily as you can a full one. Using only the amount of water you need will save water and allow you to quickly refill your bucket for consistently clean solution.
And don't forget to clean your mop heads. A clean mop should always be used between accounts to prevent cross contamination. Dipping a dirty mop into clean water will make it instantly dirty, so always have a rotation of clean mop heads on hand.
Using these quick tips you can greatly increase the effectiveness of the traditional mop and bucket while conserving water at the same time. Old fashioned method. Modern efficiency.