You can still eat healthy foods. The most important thing is for your diet to be steady. That is, you should not eat lots of foods containing vitamin K or vitamin E for a period of time, say a week, and then switch to a diet containing none of these foods. The healthy foods are actually recommended by most doctors. The Coumadin dose is then adjusted to compensate for the intake of these foods. This is the solution recommended by my doctor. The doctor at the hospital did recommend I cut out green leafy vegetables. I doubt that many doctors are this stupid.
Remember that your health is your responsibility. Always seek second or even third opinions and do research, as you are doing.
You might also include herbal blood thinners in order to reduce the requirement for Coumadin but should be very careful in dosage and consistency, and should not attempt this without working very closely with your doctor. Most herbal medicines do not have any consistency in potency, so most doctors will be leary of dealing with them.