We found a really interesting, trendy district close to Monastiraki, called Psiri. It's centred on a square, Plateia Iroon, and is a maze of little streets. It has many small taverns, and is frequented by Greeks, not tourists. We had a really nice meal (Mezedes again!) in a place whose name I could not make out, and listened to live Bouzouki music. This was the real deal, two great musicians. We were the only non-Greeks in the place, but they welcomed us warmly. Had a great time.
These are modest places, but nice.