Natural Weaponry Power covers this as does Weakness detection, and Heightened Expertise and some heightened stats (no skills, so you cover that with higher stats, like Agility.) You can build a pretty solid one. The table for "random" rolling that is labeled Skills? Give you multiple powers that cover innate training "powers."
In fairness, there's some stuff that's pretty much the same between the two editions. But some stuff is different, as well.
Martial artists have one big drawback compared to other types of characters, and one advantage that might make up for it. The big drawback is that the base chance to hit for melee attacks, compared to other powers, is fairly low. You would ideally want to have Heightened Agility and Heightened Expertise to help counteract this. The advantage is that you don't have to pay Power to make a melee attack (although you still have to pay some Power when making multiple attacks per turn), which not only means you can go longer in a fight, but because you're not spending as much Power on your attacks you have more in reserve to burn 'rolling with the punch' to soak instead of taking HP damage.
Besides Natural Weaponry (which can represent either martial arts or claws, depending on the type of character), Heightened Agility and Heightened Expertise are the two other powers/skills you probably want most as a martial artist. Heightened Defense is also highly desirable, as it gives your opponents a -4 to hit you, which is really useful if you're plowing into a bunch of mooks. Heightened Endurance and Heightened Strength are also very useful for obvious reasons. Heightened Attack is great at higher levels, but as it's dependent on character level, doesn't really pull its weight when you're low level. Weakness Detection is quite useful, but you have to spend an action studying your opponent before getting that nice bonus to hit them.
If possible and thematically appropriate for your character, see if you can justify with your GM having a weapon like a katana or nunchucks - not only will it add to your damage, but they will give you a bit of a bonus to hit as well.