One of Chicagoans' favorite things to talk about is how we talk; thanks to our rich export industry of sketch and improv comedy, the Billy Goat's line cooks and Bill Swerski's Superfans still define Chicago for much of America. We like to think it's unique, but there's another city that sounds like us. Cleveland.
There's some logic to this. I've written about this before: how it's believed that what we think of as the Chicago accent has its origins in the Northeast and moved along the Erie Canal, which connected the Atlantic Ocean to the Great Lakes by way of Albany and Buffalo, creating a flow of immigrants from northern New York to the Midwest through what's now considered the Rust Belt. Then politics and war deepened the linguistic divide: "Labov explained that locals in such areas as northern Ohio and Michigan traditionally spoke precise English because they wanted to distinguish themselves from the speakers of Southern dialects in their states—a split that seems to go back to the Civil War." (To this day, some Southerners—like myself—avoid sounding Southern when they're not in the South.)
Above is the aggregate map for the answers to Vaux's many questions, which emerges from barely noticable differences into something of a pattern. Katz also mapped the individual questions. Curiously enough, the question where you can see this pattern most clearly is this: "Do you say 'frosting' or 'icing' for the sweet spread one puts on a cake?"
Conveniently enough, it looks like a layer cake.
Another unsettlable linguistic distinction—you've probably argued about this one—also shows the pattern. "What term do you use to refer to something that is across both streets from you at an intersection (or diagonally across from you in general)?"
Southerners sometimes keep their heritage close to the belt, but you just have to ask the right linguistic question.
As to whether Chicagoans really do sound like Clevelanders, well:
IT’S ALL ABOUT STRESSING THE A’S! This one is a biggy! I get made fun of for how I say my’s A’s. Especially in words like “accent.” Make sure it’s a long A sound as if there were another vowel. Draw out that A. Ex: Aaaaaccent!
Sounds about right. Combine that with "Please, never say soda. It's POP!" and you're in the ballpark.
Update: Cleveland-expat correspondent @nocoastoffense notes that Mike Trivisonno is "the foremost example of the Cleveland accent." It's not far off our own.