It doesn't have to be too advanced, but the zombie that you're attempting to fight better be pretty decayed for you to deal head damage. I mean in Episode 5 of the game, Kenny ******** pistol whipped a zombie dead. Made it his b***h.
I'd agree with masterninjabunny. Freshly turned zombies are gonna be tough to punch thru the head in order to get to the brain, so unless you're name is Iron Fist or Shang-chi I'd steer clear of the martial arts vs. the undead.
Martial arts would be fine so long as your wearing protection. I always said that I would wear armor of some kind something like leather biker stuff with some good gloves. Get some good gloves and something to use in your hand like a knuckle duster. However a hammer or something would be easier. I have a nice spiked tomahawk. Any splatter may be risky to you rarely see in these shows the splatter causing someone to get infected. Wearing a helmet is a good idea.
Martial arts would be useful in a sense of controlled direct movement, as for solely protecting oneself - I wouldn't count on it. You want as little direct flesh contact as possible, the closer you get, the more likely you're getting bit. You could use martial arts moves to back up a weapon such as a kendo (sword) or bojutsu (staff) and do enough damage to kill a zombie.
Eh, the closer you are to them, the worst. Long distance is much easier, much safer, and much more useful. If you use punches or kicks on 1 zombie, there's probably like 30 more behind it. You'll get surrounded, get pushed to the ground, get eaten, and become what you kill.
In most martial arts you learn the value of defense and channeling your energy in ways you can harm or stop what your offender is doing without killing them. Zombies don't have pressure points or nerves, so that would pretty much outrule any attack you're using below the head and like Konpon said, it'd be pretty difficult for you to punch or kick through a skull, much less 20 or 30 of them. Unless you're Cottonmouth from Kill Bill or are skilled in Battojutsu you're not going to have much luck with zombies using only martial arts.