Most visitors to Athens will not miss Acropolis, Plaka and Syntagma, where most of the historic attractions and many museums can be found. But these central neighborhoods also offer a variety of choices for lunch or dinner. Kolonaki (for shopping) and Exarcheia (for its bohemian atmosphere) are also favorite itineraries, and also the cruise port of Piraeus (for those visiting Athens on a cruise).
Restaurants to include on your itinerary close to Acropolis would definitely be famous Dionysos, which has a fantastic view of the Acropolis and an elegant environment; cozy Italian trattoria Aglio; Olio e Peperoncino, just off Makrygianni Street, with an inspiring menu; and Strofi - opposite the Odeon of Herodes Atticus open-air theater - with its cool terrace and Greek specialties. 
View of the Acropolis — Photo courtesy of Dionysos
Strolling through Plaka - the most picturesque neighborhood in Athens - you can taste traditional Greek dishes at renowned Daphne’s favorite restaurant with visiting celebrities. Or enjoy a more casual atmosphere at Diogenes, with a view of the Lysicrates monument.
Platanos Taverna, close to the Tower of the Winds, is a good choice for value and homemade cuisine. Closer to Syntagma is Avocado, which specializes in vegetarian cuisine with wonderful healthy tastes. Right opposite is Makalo, with its innovative and interesting cuisine.
2Mazi, on the same street, is an interesting example of traditional Greek cuisine with a gourmet twist, based on the inspired knowledge of two talented chefs. Proceed to Kolonaki, where some of the best choices are Abreuvoir, for French cuisine in a cool garden; friendly Paella Barcelona, with its primary colors and Spanish specialties; or Filippou for mama's cuisine in a classic atmosphere.
In the same area, but close to the center, Seventeen is famous not only for its excellent cuisine and elegant atmosphere, but also for one of the best martinis in town, while Prytaneion, on busy Milioni pedestrian street, serves some of the best brunches.
Continuing to Exarcheia, with its bohemian bars and clubs, you should definitely try the delicious crepes at Au Grand Zinc and Cretan cuisine at Oxo Nou, both very reasonably priced. Yiantes, also in Exarcheia, specializes in dishes from all over Greece that are always prepared with fresh ingredients.
If you are visiting Athens on a cruise and want to stay close to the port, then try one of the many fish food restaurants that abound in the neighborhood, such as renowned Vassilenas, Jimmy and the Fish or Margaro for something more casual. Elegant Istioploikos on Mikrolimano and original Yperokeaneion on Hatjikyriakou Avenue are also very good choices, each with its own unique style. 
Original decor at Yperokeaneion — Photo courtesy of Yperokeaneion