The day I asked this one housecleaner of mine to clean the grout between our kitchen tiles was the day she quit. (She only lasted like 2 weeks.) She didn't do a very good job on the grout and that was probably pretty discouraging to her. But now I understand why. I didn't give her the right tools to do it.
The grout between our kitchen titles originally matched the light beige color of the tiles themselves. 10 years later, the grout looked closer to a dark shade of black than beige. No amount of mopping or even scrubbing with a toothbrush helped.
Then I got this tool off Amazon that was a spinning scrubber. It totally worked to clean the grout FAIRLY well. I was so pleased. The tool was only $50 and I used it for the hard to reach corners, the inside lip of the bathroom sink, a bunch of places it cleaned decently.
BUT THEN, Dave started researching how to clean tile grout and he found something EVEN BETTER and EVEN CHEAPER. All he had to get was a drill bit cleaning brush. It attached to his drill, and he went to town on the grout that I had already cleaned once (!). The drill was able to get at it way better. It was faster, you could put more pressure on it, and it was easier to control. I highly recommend getting a drill bit cleaning brush set if you don't already have one. That is not an Amazon affiliate link right there. It's a sheer act of altruism. Your grout will thank you and your housecleaner won't leave you.
Warning: the below video is quite graphic. In it I out myself on how filthy our grout really was...while moaning in a slightly disturbing way.
This is what the spinning scrubber did. It was decent. But the drill got the grout color even lighter. We had completely forgotten what the floor used to look like!
BTW all I'm using as a cleaning solution is the Degreaser from our hypochlorous acid maker machine. I spray it into the area before the drill does its magic. Then wipe up the brown water with a microfiber cloth.