how do you clean green deposits from stainless steel eating utensils?

Get some Barkeeper's Friend. It's a powdered cleanser that works great on metal. It's abrasive enough to do a good job but not abrasive enough that it scratches. Scratching is the problem with the Brillo or SOS pad; they will scratch.

As to what's causing it, either your dishwasher isn't cleaning well or it's a chemical reaction between your dishwasher detergent and something in your water.

You could try baking soda and water in a paste and scrub them with that but it's probably not gritty enough to get it off. Get the Barkeeper's; it's cheap and it works.

Get a soap filled steel wool pad--like SOS or Brillo, and go over the eating utensils to remove the green deposits. Then wash the utensils in hot soapy water and rinse thoroughly. then dry each piece with a clean towel, and lay them out on some paper towels for about an hour, cover with paper towels as well. Check each piece before returning them to the drawer. It could be that they are plated with something. I know my mom had some silver plate eating utensils that were silver over copper, and the copper started to come through after many years of wear.

I would suggest that you not put your eating utensils in the dishwasher. If they are plated, that will wear away the top layer easily as dishwasher detergents are quite harsh.

it may just be cross comtaminating

i use quickleen-s for cleaning copper it is safe and really fast.

how about not trying to put anything on it that you wouldn't want in your system?