We went to the Psaras Taverna on the recommendation of a Greek friend who said, "Tourists eat there, but Greeks also eat there." Our meal of a Greek Salad, Sea Brean and Lamb, was very good. When we asked what the fish of the day was, the waiter ushered us into the kitchen to see the options. The service was excellent. Dining was outdoors on a patio, indoors, or on a covered porch. It was about a 10 minute walk from the Acropolis Museum and a little hard to find, but worth a little bit of a search.
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