I am a L.A. transplant in Chicago but never have been to Japan so take that into consideration in my response...
When I think of 'traditional' of sushi joints - in terms of the overall experience (not just quality of fish), I think of izakaya style restaurants - which are essentially places to go drinking and mop up the sake and beer with some food. Mostly sushi, sashimi and a few other items. They typically offer omakase which is a chef's choice (not a fancy prix fixe, but kind of 'hey, just send me whats good today'.)
At one of these 'traditional' places the desired location is to sit at the sushi bar, and many will wait for the bar while tables go unseated. At the sushi bar, you should deal directly with the chefs behind the bar and either just say 'omakase' or start ordering what you want through them. Also, if you are drinking, you should buy the chef a beer / sake and they will certainly take it from you

If do you order Omakase, it should come one to two items at a time over the course of an hour or two, not all at once on a plate, boat, wooden board etc.
From the my experience in L.A., Honolulu and Manhattan the best Izakaya joints are loud busy places, with unfussy decor, a natural wood sushi bar, perhaps a TV in the background and for the larger places, a tatami room (sit on the floor / low table / take off your shoes) in the back.
Sooo..... after three years here in Chi town...
My answer to you is that there are NO traditional sushi restaurants in the city at all. Ginza is about as close as it gets, Katsu is has the best sushi / fish in town but it is less traditional (as I define it) than Ginza.
In fact, from what I understand from Katsu-san only Ginza, Katsu, Kuni's and one or two other sushi joints are actually owned by Japanese families. Note: I am not saying that any other nationality could not successfully own a 'traditional' sushi joint, but being japanese would certainly be an advantage.
Don't worry - There are plenty of pretty good sushi joints here. Nothing compared to the coasts where the ocean fish comes into port and demand is much higher. Tank, Katsu, Ginza, Tanoshi, Kuni's are my go to's. Just be sure wherever you go try to order the sushi chef a beer!
There is no accounting for taste!