Mack, the problem that you have is not uncommon and can usually be fixed by the installation of a water hammer arrestor. The sudden shut off of the water supply to the dish washer causes the pipes to hammer or shuder as you have described. There are a number of model and types available and they should be fitted by a licenced plumber. I do not know where you live but if it is in Australia then use the good quality plumbers that are availble with the system.
I have the same problem in my home. When the washing maching kicks on it seems to set off a toilet stool at the end of the hall (other end of house) and then the copper pipes rattle like a fright train for anywhere from a few seconds to a few minutes. At the same time the stool valve seems to kick on and let more water into the tank. Some times this is everytime the valve opens in the automatic washer and other times it goes away for days and then starts again. What have you found out from others? Would you mind sharing? Have you found the solution? REGARDS!