We once had a house on septic. The dishwasher and sink(s) water was piped out of the house separately into a grey water discharge area well away from the septic field. This discharge area was simply a large pit in the ground and filled with 3/4 inch crush rock and sand.

It would be fairly easy to have an existing dishwasher drain using this method, however for existing sink plumbing, it may not be feasible. Also not all locales support grey water discharge so you will need to check local laws.
_________________________
Earth and sky, woods and fields, lakes and rivers, the mountain and the sea, are excellent schoolmasters, and teach some of us more than we can ever learn from books.

John Lubbock