My stepdad was put on coumadin during a hospital stay some months ago for post-surgical blood clots, so I met with the staff dietitian about his nutritional needs. (Eyeroll. I apologize if any of you are hospital dietitians, but I did not enjoy the conversation.)
The short answer is none - iceberg lettuce is about as exotic a green as they can eat. The basic reasoning is that they need to keep their vitamin K level very stable and consistent. For most people who eat SAD, that level is low, and so it should stay low.
However, if someone was eating greens before starting the medication, then they should keep eating the same amount and types. Don't change the intake of greens, in other words. My stepdad was in the habit of eating a small green salad before dinner, so the dietitian advised him to continue that so his levels wouldn't drop.