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.