After experimenting with various types of cleaners, I found the best way to clean white resin patio furniture that has baked in dirt from many summers of sun exposure and garden soil. I use a combination of Regular Vim and then the Vim Gel Bleach. This seems to produce the most cosmetically pleasing results.
I start by hosing the chair down in my driveway, and then scrub it with a rough bristle brush using the Regular Vim. Sort of like an initial chemical peel. Then I wet the chair again and apply, with a sponge, a light coating of the Vim Gel Bleach, and then let it sit for a few minutes (not too long, though, maybe 5 minutes at the most.) This seems to take off the second flaky resin layer that really traps the dirt, particularly if the furniture has grooves or texture to it. I then scrub it some more with the bristle brush and hose it down.
I have some great before and after, as well as in progress pictures of some very grubby chairs of my own that I did using this method, and also some that I found in the garbage that matched my original chairs, and which responded equally well to the treatment. It's pretty dramatic to see the difference. The chairs look almost new after I finished, though I'm sure sloughing off some of the baked on resin powder and residue as I scrubbed might have made the chairs more vulnerable to further sun damage. However, now I take care of them and keep them out of the sun when they're not in use.
I was also considering using the Krylon spray paint I bought that is made just for plastic patio furniture. However, since the cleanup went so well, I put the cans aside for the day when the chairs are too scrubbed to death to do anymore :) Also, the colours of the Krylon are somewhat limited, though practical: White, Beige, Hunter Green. There might be more, but I haven't checked for a while.