Middle Eastern cuisine has a lot of meat dishes but not nearly so much dairy, and the vegetarian dishes tend to also be vegan: falafel, hummus, Jerusalem salad, Persian salad, maklouba, baba ghannoush.
South Asian cuisine tends to be vegetarian and the dairy is usually ghee (sub in olive oil or another liquid vegetable fat). 90% of the recipes are already vegan-friendly: Chana masala, tarka dal, aloo gobi are my favorites. And if you do come across a meat dish, sub in green jackfruit and you'll have a good time.