1) Dirty Water Dog - Yes, it may give you a disease, but it is delicious. And the way to eat it is with mustard. Just mustard. NO KETCHUP. Maybe get a pretzel there while you're at it. And have that with just mustard. But stay away from the knishes.
2) The Gigantic Every Food on Earth Available Deli Place - These places are crazy. The place to go if you feeling like having a bowl of gumbo, a turkey wrap, stir fried veggies and krab and a california roll on the side. You can go to the sandwich counter, the tossed salad counter, the hot regular grill, the mongolian grill, the cold salad bar, the hot salad bar, the sushi cooler, the pizza counter, and the smoothie counter. And probably more. So none of the food is fantastic, but most of it is not bad, and the variety of stuff available is wild. A good one to visit (though sort of out of the way)is Deli Grandville on 10th ave. Stop by after shopping at B&H.
3) Popeye's Fried Chicken - OK, I know, not New York. But I used to live in Lousiana and discovered its joys there. Whenever I am needing a nice does of greasy deliciousness for lunch this is my standby. And I think if you come to the US, you need to try some fried chicken, and Popeye's is just the best there is, especially in New York. And make sure to get a biscuit. Map to New York Locations.
4) Pizza Slice Place - I place these in a different category than the "full pie" pizza places. Pizza generally is so so (so so for New York, great for much of rest of country), but quick and cheap. By the way, most of them are named Ray's, but there is only one real Ray's, the rest are imposters. A good one is Maffei's on sixth ave.
5) Full Pie Pizza Place - Now this is strongly debated thing in New York. I won't even get into it except to say that I used to live in Brooklyn Heights, so Grimaldi's is tops for me. If you are an out of towner, you need to take the train the Brooklyn, eat at Grimaldi's, then walk back over the brooklyn bridge. Such a great afternoon and so few visitor's do it. For the real pizza enthusiast, here issome further reading.
