Eat at God's Restaurant, just a few blocks south of the Acropolis (on the other side of the "hill" from the Plaka neighborhood). 

www.alexander-restaurant.com

http://www.varoulko.gr/

www.dionysoszonars.gr

www.48therestaurant.com

www.spondi.gr

http://www.daphnesrestaurant.gr

www.edodi.gr

psaras-taverna.gr

www.fishcafe.gr

Another fabulous place to eat is the metropolitan cafe restaurant in Monistiriaki square - very good service, a beautiful little square and very reasonably priced.

There is also Roma Italia,'Tilo's,Steak House, Thermion and Marina

If it's smaller cafes you're looking for you will have an extremely wide choice. The glitzy beach cafes are all along tram route 3 and the more city based ones are on almost every street. It was refreshing to see very few chain cafe's here and in Omonia Square there is an amazing deli just around the corner from the coffee house. (Go down September 3rd street just a few metres and it's opposite the first set of pedestrian crossing lights) Do try the iced coffee wherever you go, it's very unusual.

  [  wouldn't suggest Omonia square and the around area. It is a bit dangerous....]

Across the road from the Novotel Hotel in the north of the city are some of the local cafes where a tourist is rarely seen. This is where genuine Greek culture is and well worth a visit for a vodka or bar meal. 

If you are looking for a wider choice of deli's and cafe's then September 3rd street is off Omonia Square and well worth a walk along.

Finally, a quick note about prices. As you can imagine, the more central you are, the more you will pay for food. If you are self catering in Athens, use the Eurospar supermarket on the tram routes (It's the stop before all the trams turn onto the coastline seafront). Also, you will find on the same route an amazing locals market where all manner of goods are being sold at very low prices - just don't forget about your baggage allowance!

Whatever happens, you will not go hungry in Athens!