As far as I know, Eatzi's failed miserably when they tried to come to New York about five years ago. They were overlooked in a city whose takeout options are, well, vast. In Houston, Dallas, and Rockville (more suburban places where takeout options are pretty much limited to the big chains and the upscale local minichains) Eatzi's flourishes. Chicago's situation, where, especially on the North Side, takeout options are plentiful and don't often involve getting in a car, is more like New York's. Eatzi's would probably stand a fair chance in a North Shore suburb like Evanston, Northbrook, or Highland Park, or in a boomburb like Naperville. Yeah, Naperville. However, Chicago is probably pretty likely not to notice them.
Anomaly: About five years ago, Memphis had its own equivalent of Eatzi's, called Cheffie's, in a well-off part of East Memphis similar to Dallas' Oak Lawn or Houston's River Oaks. Cheffie's lasted about a year before tanking, despite the lack of takeout options beyond barbecue, pizza, barbecue pizza, or mediocre Chinese in the area.
Now if anyone knows Pete & Sam's Italian Restaurant, on Park at Getwell in Memphis, THAT's something I wouldn't mind seeing here in Chicago--it would make out gangbusters on the Southwest Side or in the near southwest suburbs. Funeral-parlor decor, waitstaff older than dirt, killer steak sandwiches, well-done thin-crust pizza (including that Memphis favorite, barbecue pizza), and toasted ravioli to rival St. Louis... can't you just see it somewhere around 95th and Pulaski (although I think it's called Crawford over there)?