I had to put down 2 months deposit for a six-month rental, even though I was employed. It seemed that owners in Illinois are wary if you don't have a credit history, so if you are from a country where you can carry that across somehow it would be good. I wasn't able to move mine across, so had to start from scratch again. Even letters of position from your bank don't seem to carry much weight.
It seems credit scores and having a permanent address are important for many things in the US. Maybe folks here will have some advice on how to build the credit scores up for you. I got AMEX to transfer my credit card over, which helped. Credit unions might be a good place to start with. Be careful not to let people "ping" your credit too much, even when they say it would be a soft and not a hard ping.
Even if you don't plan to drive, I'd encourage you to get a driving licence. It is a very handy ID, on top of your GC.