I have been to The Greek Islands since the time in the very early seventies where they were a humble steam table kind of place on Jackson. I tried all the other places in Greek town. Once I became addicted to the Gardens of Plaka, then to Santorini's, then farther North to the the Athenian Room on Webster.
I also tried Avli in Winnetka, Mykonos in another Northwestern suburb, and others...
I had dinner only once at Taxim and was very disappointed by the food, that is pseudo-original.
Grecian Psistaria on Devon is fine since it was taken over by alumni from Greek Islands.
But you know what? After 42 years of patronage, I always go back to the Greek Islands. To me it is the most reliable, welcoming, comforting, and authentic Greek restaurant in Chicago. From sea bass to lamb and artichokes, everything is satisfying. And the human approach from the management and waiting staff is the best.
But Avli has the best Greek wine list by far, and the patio is great during the summer season.