I lived in Lincoln Park about 5 years ago with my GSD, but I owned my property.
Have you looked into using a leasing company and telling them specifically what you are looking for with a GSD? I agree a private owner would be more receptive than a managment company.
The links below are a few sites you can visit and see if they can help. I like the last one, dogfriendly.com. It lists a whole bunch of info.
Good luck with your search!
Sites that List Pet-Friendly Apartments - Petfinder
DogFriendly.com's City Guide: Chicago, Illinois