Apart from leather hinges that were used even when iron was common (but not so common as to be cheap), there existed the different mounting of the door. I have heard about its use from my grandfather, when I asked for explanation about what is the "heel" of the door in Russian fairy-tales.
The whole side of the door frame worked as a huge hinge - this side was made of a bit longer piece of wood, that fitted into shallow holes in the appropriate places in the outer frame. The sticking out pieces were called "heels". It is mentioned by several Russian fairy-tales, when a Door asks a hero to grease her heels. (the good hero does and gets some help).
Obviously, the very thick and heavy wooden doors of the old Russian houses simply could not be held by leather strips. (The boards were extremely expensive). And the leather had to be attached to the door, too.