Dumping grey water into a tree bog.

I got this from the website of Oasis design. He goes into some detail, but I thought this statement sums it up pretty well.

"Health risk is extremely low. There are over eight million greywater systems in the US, and in the 60 years that the Center for Disease control has been keeping records, There has not been one documented case of greywater transmitted illness in the US. "

For me, using grey water, is mostly about using water that would otherwise be wasted. I don't think I'm going to be that one person in a trillion who poisons themselves with bath water.

I have lived in two different houses that simply ran a pipe downhill, to a forested area and allowed the water to percolate there, amongst the trees. In both cases, it was distant and downhill from, the well. We don't have a shortage of water here, so this was just expedient disposal. A rake was sometimes used, to incorporate bits of food waste, into the soil. Just a few times a year. Worms hung around at the outlet.

One of the houses had a wooden box, covering the pipe outlet. This was meant to keep wildlife from feeding on the pipe effluent.

I could see this being a problem, if it were adjacent to a body of water, where fish could be harmed by soaps & detergents. Those trees grew really well.

 I like the idea of using only gravity, to move the water to the percolation field. No settling tanks, no screen and definitely no pump.
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In areas where there is no shortage of water, do you see anything wrong with this sort of disposal system?

I don't see anything wrong with the system you describe. You're right, it would be a different situation if it was close to a drinking water well, creek, river etc. because then the nutrients in greywater would be pollutants and cause algae to grow. If the area was really boggy, with a high water table, then greywater could get into the groundwater, which is not something you want. The water will be purified as it travels through soil, so you should have a few feet of separation between the discharge and groundwater level.

some trees don't like alkaline water, some do and some trees don't like wet feet.

I should add to my previous post, that the best type of ecological disposal system for greywater would be a constructed wetland or self-contained planted area (like an evapo-transpiration bed). This method allows greywater to be used, and used-up, without any impact on the natural environment, and is important to use if the site is near any water ways.