Of course you can have black tea, there's no Vit K in it, or not enough to matter. You can have green tea, too, if you drink it on a regular basis, though you may have to adjust your dosage of coumadin upwards if what you drink actually has lots of vit K in it. Some do not.
There is nothing you can't have, including soy and green leafy veggies. Dose the diet, don't diet the dose, and if you are knowingly adding in extra vitamin K sources to your diet (green teas, lots of spinach or brussels sprouts, etc.), or otherwise changing things around, be sure to test weekly. The only thing you need be concerned about is sudden splurges or sudden withdrawal from your vitamin k sources or activities, and then it just means adjustments to dosage if you test regularly.
Sometimes they fill us full of nonsense in the hospital, and with the written materials, because not very many medical providers really understand managing coumadin/warfarin. The key is reasonable consistency. We have a fair number of vegetarians/vegans on coumadin, and you don't see them starving to death. In fact, if you try to eliminate Vit K from your diet, or minimize it, you will have a much harder time stabilizing your INR.
Go visit Al Lodwick's site. See the stickies up top of this forum! Eat!