White horses don't necessarily get any dirtier than a brown horse, they just show it more. I have a 50/50 paint...but his white is on his legs, belly, etc. primarily. His legs have a permanent tint to them, but I usually use Quik Silver (whitening shampoo, that is actually purple) a few times a year. I keep him in a fly sheet for both the pink skin, which can sunburn, and to keep some of the dirt off of him. I would definitely get a fly sheet with UV protection for summer months, and a good turnout blanket for the rest of the time. That will keep some of the dirt off of your horse.
One thing I've noticed is that my horse, and 2 others with white legs are much more prone to scratches on their legs. Mine is just getting over it. It didn't matter how hard I tried to keep him from getting them, being turned out in the northwest/wet climate it was just inevitable. I just keep cleaning his legs and treating it.