You must rest at some point, the zombies never do.
Let's assume a shambling zombie horde travels at two miles an hour. To get a full eight hours sleep you must get sixteen miles ahead of the horde, assuming you can go from running to sleeping instantly. What's more likely is that you'll need around twenty miles head start.
The Roman army generally traveled eight hours a day, about thirty miles, which is quite an effort after the first few days. You don't really get the time to properly rest your legs. The horde will be traveling around fifty miles a day. You think you can do around fifty miles a day indefinitely to stay ahead of the horde?
The fear from zombies is not that they're dangerous individually, it's that they're tireless and relentless. They never get tired, never get bored, never stop. Not really something that can be done easily in film and something that is even harder to make into a fun game.