Using a vacuum cleaner on tile floors before you mop, will keep from making mud pies in the grout!

Use a regular string mop to clean floors. The strings will swish around in and over the grout getting it clean instead of shoving dirty water into the grout when using a sponge mop or other type.

For heavy duty cleaning the tile/grout, make a hot solution of Finish Power Ball Dishwasher Cleaner. Wet floor good with solution and wait 30 minutes before scrubbing/rinsing. If floor drys out in areas, wet again with solution. Number of power ball tablets will depend on how much solution you mix up. About 4 tablets to a gallon of hot water might be what you need. Use a grout brush on the grout. Janitor supply places will have a beveled type grout brush that will fit on a long broom handle (save your back).

Get the steam cleaner. Steam cleaning is the easiest way to clean ceramic floors, IMO. You will need to vacuum the floor first to remove the dog fur.