Here's a post I wrote a while back comparing different grout cleaning methods: https://www.curbly.com/2254-how-to-clean-tile-grout
I have tried many recommended solutions and found that the one that really works for me since I have a light grey grout is Clorox Toilet Bowl Cleaner. It is a gel and I put a dab in the corner of four tiles, spread it along the grout lines with a tooth brush, wait 10-15 minutes, brush again and then rinse off with a sponge dipped in vinegar and water. The vinegar solution removes all of the soap residue. I then steam clean or damp mop the whole floor to be sure to remove any further residue. Do not get the cleaner on your wood floors.
I recommend a cleaner called Oxi Clean Max Force. I have been using this product for almost 2 years and been happy with results. I scrub it into the grout with an old toothbrush. Wait 15 minutes and then wipe it all up with a damp cloth.