the easiest thing would be to go to walmart and stock up on 1000$ money orders paying a 70 cents fee for each. your landlord would almost certainly accept money orders, but double check first just in case. i say stock up because you can't get a money order for more than 1000$ and they usually won't let you buy more than 3 per day.
alternatively, you can probably open a bank account using your ssn and your passport. look for any bank offering "free" checking, and they should be able to give you a few "starter" checks on the spot when you open the account. in any case, they can certainly get you a cashier's check for free or a small fee.
side note: if you want to shop around for a checking account, look for a bank or credit union offering a "kasasa" account.