Although it took me six months to figure this out, the technique is extremely simple. First, I decided not to use any water from the kitchen for this gray water system, because of the difficulty of filtering out food particles. This system exclusively uses water from our clothes washing machine, which does contain a lot of lint, but that is easy to filter out. NOTE THAT, FOR HEALTH REASONS, IT IS NOT RECOMMENDED TO WATER VEGETABLES WITH GRAY WATER (fruit trees are OK though).
Drip irrigation systems water individual plants with little parts called "emitters" that allow the water to drip out. They are specially designed to not clog with roots or debris, but they require a fair amount of water pressure to operate and everything I have read about gray water systems warns that emitters will become clogged if you try and put gray water through them. However, there is another part, frequently used in drip irrigation systems that is the key to my system. It's called a 'barb x barb" connector and you can see a picture of one here
(I'm not trying to endorse this vendor, it's just the first photo I found on the web). The barb x barb is made to connect a 1/4" diameter extension hose to a larger main line. In my system, I just put one of these barb x barbs next to each plant that will be watered. The first photo below shows one of these "barb emitters" in action.
Each barb x barb has a hole that's maybe 1/16" in diameter, and this is large enough so that it won't clog (at least they haven't clogged after the first few months of use last summer and fall). In my first version of the system, I put the barb emitters at a slight upward angle, like in the photo. There were two problems with this: First, in some parts of the terrain, the pressure became low enough so that having the barbed emitter pointing up led to low or no water flow. Also, one of the nicest things about drip irrigation systems is that it delivers the water underground, so you get less evaporation. Putting the barb emitters straight down, sticking into the soil, would help the gravity/pressure problem and also the evaporation problem. However, just sticking the barb emitter into the soil could lead to clogging with particles or roots.
So, my second version goes as follows (see photos 2-5 below): I insert the barb x barb into the hose pointing straight down, then I wrap the middle section of the barb x barb with adhesive foam weather stripping. I then use a wooden dowel to poke a hole in the soil just under the barb emitter. I take a 3" length of irrigation tubing, fit it snugly over the weather stripping, and insert the tube into the hole in the ground. That way, the barb emitter has an empty space below it that is kept open by the tubing. I have had some of these in place since last fall and soon I'll check them to see if they are still working. Overall, I think the second version is a better idea, but there's something really gratifying about seeing the water squirting out of the above-ground barb emitters.
***UPDATE - the weatherstripping didn't hold up very well over the winter, so I'll need to find another way to plug up the hole. For now, I'm just leaving them - I have decided that pointing the x barb x straight down doesn't work in the long term, because the hole always fills up with soil, even when I put a piece of irrigation tubing inside the hole. Therefore, I recommend just having these "emitters" at an upward angle, as shown in the first photo.
Sometimes I don't have the main irrigation tube very near a plant. In those cases, I just insert a barb emitter nearby, and then insert a piece of 1/4' tubing to bring the water flow right to the plant (which is what barb x barbs are meant to do).