Using google maps and some sketching, I have a better overhead view to show. On a side note, it looks like google got new (at least to me) satellite data from this summer. And the new 3d features are pretty cool as well. But I digressed.
The sketch drawing is as follows. The yellow line is a berm along the property line as the neighbor above is up stream of us. Without that berm to redirect the water, our house would be flooded with their run off.
The blue lines indicate how the water currently drains (except for the color photo, which is in red).
The black arrows are the downspouts, two on either end, and one in the middle. The two flows going towards the back of the property line, are both in low areas, meant for that drainage. I'd say they were swales, albeit shallow ones. Where the pad for the shed is, and where the garden is located, are on the high ground.
Overall I'm not trying to make any major changes to the flow of water. It moves well away from the house, just a few low lying areas that take forever to dry out. We get a lot of mosquitoes, and that is one of the main reasons I'd like to fix it. The drainage is a tricky issue, as the house was built a tad low. They really should have raised the foundation another foot or so when the house was built 40 years ago, would have made this a lot easier to grade and drain.
The main problem area is the upper right, the water flows down the back of the two upstream properties and then will just pool behind neighbor 3's shed. The slope drops about 4 feet over 15 feet where upstream of the large red circle. The water over time has been making a mini grand canyon between the two upstream properties.
I'd like to raise the ground by the side of the hosue an inch or so, and keep the water flowing to the front and back as indicated, then just add in a bit of fill in those low spots. The termination point up near the shed pad, upper right, is actually a bit high, only say an inch or two, that would need to be knocked down. That'll allow the side flow to move into the back drainage lane.
I'd then fill in the big low spot (circle in red on the photograph), but keep it sloping down stream. Just enough to stop the pooling and not force any into the back neighbors shed/back yard.
I like woodchopper's idea of the 4" corrugated pipe from the upper right hand corner of the house to the back of the property line. It'll have to be shallow though, as there isn't much of a slope, especially if I have to raise that big low spot. The other issue is the tree roots, not sure I could get very deep without hitting them.