Buy a stack of white cotton washclothes at Costco for 9.99. They are really nice and you get 20 of them!
Get a container of warm water and add to it a *little* plain white vinegar. Change water solution and cloth often when it gets dirty.
Get down on your hands and knees and go to town like ATW's clip showed

. They thing is to not let any water stand on floors. Sometimes I dry them with another cloth.
Then throw the whole load in the washer and bleach them. No Downey of course.
If you stay on top of it...there should never be a need for deep cleaning like this. Just spot cleaning and dusting as you go.
HTH!