The theoretically best technique (IMO) is to dump the rice in a tub of water and let it sit for 30+ min. That starts the hydration without wasting heat (thus a faster cook). Then I grab a handfuls and rub the grains together to wash mechanically. I know it’s working because the water gets quite cloudy on every cycle. After dumping and repeating 3—4 times, the water is still a dirty color.
How many times does it take?
If you don’t wash brown rice at all, there is a nasty ring of mud around the pot at the level the rice expands to. If you wash ~4 cycles, there is still a faint ring of mud (or so it seems).
I’m way too lazy to do 10 wash cycles, or whatever is needed. I used to put the rice in a strainer and run it under the faucet for a while. But I think that wastes water and you cannot readily see any indication of when the washing is done.
The other problem: people wash their rice not just to get the mud out but also to get the starches out. I want the starches. I’m not afraid of getting fat.
I wash 3 times in cold water utilizing a strainer and a bowl. The rice goes in the strainer, and the strainer in a bowl. Cold tap water is turned on and the bowl filled and the rice is covered with water. Now I agitate the submerged rice with my fingers for a few seconds, although agitation with a large serving spoon would also be fine. After sufficient agitation, I lift the strainer with the rice, let it drain until it has a slow drip, then dump the bowl of water. The strainer of rice is put back in the now empty bowl and is ready to be filled again with cold tap water. I repeat this 3 times.
The final rinse is done in boiling-hot water for 1 minute before the rice is put in a second pot of boiling-hot water for the cook.
It is possible the rice I buy is cleaner than the rice you buy. One step you could try with the mud problem is one of the cold rinses could be acidified with a little vinegar to lower the pH of the rinse water. Vinegar at 1% of the water weight is somewhere near a usable amount.
Good luck.