Actually, plenty of cultures have explicit definitions for what is
right versus what is
wrong. They are called
laws, and are generally founded upon
abstract principles or
ideals.
Also, in order to 'not believe in right and wrong in the traditional sense', one has to implicitly acknowledge that, somewhere, there is a 'traditional sense', however vague.
Generally, human cultures are themselves formed upon little more than overlapping individual conceptions of
right and of
wrong; that's what the
Rousseauian social contract basically states.
However, one does have the right to violate these implicit contracts with his or her neighbors on principle; this is called
Civil Disobedience, which was practiced by
Ghandi and
Martin Luther King, Jr.