You can try chuck eye, rib eye, skirt, flank, short rib or thin sliced belly. I personally don't care for NY and tenderloin for yakiniku but some people like it.
For the fattier pieces id just use shio (salt)...but for leaner cuts I do a ponzu marinade with 1/3 soy, 1/3 mirin, 1/3 yuzu only for about 30 min to an hour.
Wagyu, being the breed of beef, has many different qualities. For yakiniku, top qualities like A5 isn't really necessary (unless you are trying to burn through money), I'd go for just visibly marbled pieces.