I use a Bona mop which is similar to the U-Line in your link. it came with dusting pads and a microfiber hard floor pad. I don't use a bucket; I get it wet in the sink, wring it out but not completely by hand. Using a spray bottle and cleaner solution, I lightly spray directly on the floor then mop with the damp mop pad. I usually rinse it out once when I'm about half way through the kitchen and continue. I like that the pad is slightly longer than the plastic mop head so the pad can be pushed up against the edges along the cabinets.

Is it ceramic tile? Or some sort of vinyl or laminate tile? If it is ceramic tile, the best way to clean it IMO is to vacuum up the dirt and then use a steam cleaner on the tiles.

I agree with you that there doesn't seem to be any way to squeeze out the excess water if you were to dunk that mop in the bucket.

Thanks for the replies.

The tile is ceramic.

What kind of steam cleaner would you recommend? There are some articles that say steam cleaners are not that great for the durability of the flooring (at least the grout in the case of ceramic tiles).

http://www.empiretoday.com/Blog/February-2014/steam-mops

Are you saying, graywings, that you only use the steamer for grout, or do you use it for tile cleaning and then have a special feature on the steamer for grout? Would you mind providing the brand and model # of yours?

I use the steamer to clean the tiles. It cleans the grout (somewhat) at the same time, but a true grout cleaning is handled differently. I have an older model steamer by Vapor Clean.

I have a spin mop and just use hot water.