Veselka's a little better than Yaffa, which can be a little scary, hygiene and food-quality-wise. But a couple of doors down from Yaffa there's a great Middle Eastern place, Cafe Mogador, which has some of the best iced tea on the planet.
I haven't finished listening to the podcast, so I don't know exactly what you're looking for, but I can weigh in once I'm done. There are a lot of crappy overpriced tourist traps but also quite a few gems in the general area everyone seems to be talking about. I'd say stay away from Around the Clock, Dallas BBQ, and what the hell is that sort-of-Japanese place that's a mini-chain around NYU? I forget. But you'll know it because it will smell like one of those rubber bar mats that's been soaked in a decade's worth of beer.
I hear the pizza from CBGB's is pretty good if you can snag some. One of those guys around Astor Place might be selling some of it vintage on a blanket.
Apropos of nothing, I know the owners of Grey Dog but haven't talked to them in years. In the 90s, we were cater-waiters together (they used to be actors) and they hired my old college roommate Mike to manage the place for the first 3 or 4 years of its existence. I used to lurk there during his shifts, drinking free refills of coffee and writing. It's kind of shocking that the place used to be empty enough that you could do that (this was like 1996 or so).