There are several ways to dispose of an old dishwasher.
1st method is put the dishwasher out for bulk trash. Many counties do not offer bulk trash pickup. You pay for it in your taxes.
2nd method is have a junk pick up service do the job and pick up the old dishwasher for a few bucks.
3rd method is breaking down the dishwasher into pieces and taking it to scrap yard and actually making money. Of course this method is not 100% available to everyone. Not many people know or want to disassemble a dishwasher, piece it out, and separate the parts for recycling.
Some scrapyards will take a dishwasher whole and give you a wholesale price though.