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 don’t wash it at all, but I do understand what you mean by the mud. I use brown basmati if that makes a difference