Thision - Thiseio, Athens
The Thissio neighbourhood in Athens is very interesting and pleasant to visit: it's located south-west of Monastiraki and it is famousfor it numerous cafés and bars. Moreover from there you can have excellent views over the Acropolis.
The best thing about this area is that it is not as touristy as the Plaka, and doesn't have shops selling tourist "crap"... the wonderful pedestrianised Apostolou Pavlou is only home to places where you can eat and drink, not shop - hence there's more locals than tourists.
I recommend this area for sightseeing, too - as you can find there a few interesting historical monuments: the ancient Agora, the National Observatory and the cemetery of Ancient Athens - as well as some interesting churches.