SFNative

We did this by using a service for sabbatical homes. Admittedly, I am in a college town, but Chicago should have some stock of this. Landlords there are looking for people to take places on short leases, and are, obviously, used to foreigners and visitors so may not automatically ask for credit checks or other paperwork that a new arrival would not have. The places are usually furnished, so if you're coming over with lots of furniture, that may require some negotiation.

Look at sabbaticalhomes.com. We also looked at AirBnB sublets, and the only reason we didn't rent something from there was pricing. In the end, we found something off the university's own board for visitor housing.

I should add we are not affiliated with the local big university in any way, just got lucky that we found something through their site.