Over the years, I've heard several different ways that people deal with silt when rafting Grand Canyon. I'm not talking about drinking water. On my Grand Canyon trip, we were able to get clear water from springs and side streams to run through our filter for drinking. The issue is dishwashing and bathing.
Before our trip, the best (and not very good) idea was to scoop as many buckets as you had full of water and wait for the silt to settle out. We were also told that adding some alum to the water would make the silt settle faster. We used this technique for the first couple of days, never coming close to getting the water clear. We boiled it for sanitation, but it still wasn't pretty.
Here is what we found that worked really well. The green bag is a filter from a central vacuum system. I've used it as a camp shower for years, hanging it from the tripod and pouring warm water into it. That would produce a nice, steady flow that you could stand under and shower.
Well, we noticed that the shower water was clear, even though the water we were heating up was cloudy, even after hours of settling. So we hung the filter down low and poured some water straight from the river into it. What came out the bottom was almost perfectly clear! For the rest of our trip, we had clear dishwater.
This really works, and it's cheap. Any vacuum cleaner store will have these cloth filter bags. Give it a try!