Specifically, I find Yerevan very interesting. If someone from 2,000 years ago stood there today, they'd certainly realize it was the same city: the same Mount Ararat towering over the city, the same river Hrazdan running by the city, the name is pronounced very similar (pronounced Erebuni vs. now pronounced Erevan.) If you traveled a bit forward in time, to 400-something CE, then a modern inhabitant of Yerevan could even read the ancient writing in the city!
Compare that with cities like Puning, China (to quote Wikipedia: "The history of human settlement in Puning can be traced back to the Neolithic Age".) Rich history, but I guarantee that someone from 2,000 years ago would have no idea where they are. (That's certainly not all of China, though, many places haven't experienced that much change.)
Anyone know more of these cities, where it'd be obvious to the ancient inhabitants that it was the same city?