More and more regular people check people's credit history so there are two options: either get a Credit Card and start working on your history(which is something I would suggest, since you will need it eventually and if you have money for an apartment you will probably have money for a CC) or say that you are a foreign student or something and you only have an ITIN (which you should get).

If some regular person asks for a copy of your passport just give him a copy of your ID page, if he asks for your visa just look for a different place. Once somebody asks for any, and I mean really mean any, immigration document you can immediately assume that those precautions are in place to prevent undocumented immigrants from renting that place. So don't try to get around it somehow, because if they find out it might hurt you a lot.