A favorite of mine is Brio (137 east 61st street). The first time we went was on a recommendation from a friend and we have gone back many times since. Not terribly expensive for New York, but not exactly cheap either. Never had awful service, but sometimes it could be better. However, the food is always delicious.
Another one is Grifone. Its on 244 E 46th St near the UN. Havent gone there as much as Brio, but still always good. The reviews are correct - Once you go a lot they begin to know you and it truly is like family. A fairly small place and very old fashioned.
While it may not be in NYC, if anyone ever out on the Island....In Westbury, on the corner of Post Ave and Old Country Rd is Cafe Baci. We've been going there for as long as I can remember and it is ALWAYS good. Be ready to wait though...on a Friday or Saturday be ready for 90 mins...but its worth it.