There have been cultural backlashes against elements of traditional culture for as long as there has been culture to rebel against (look at Ovid and the Roman satirists for a great early example), however that is far too massive a scope, so I'll stick to modern American counter culture. The counter culture of 20th century America is, more than anything else, built upon the cross pollination of white and black culture. Prior to the Harlem Rennaissance and the birth of Jazz, Black culture was seen as universally inferior to white culture, and very few people during that time period would have claimed otherwise. With the rise of black culture, especially in Harlem, mainly during the 1920's, there was suddenly a number of white youths who found this culture appealing, often moreso than their own culture. These white's began to adopt elements of black culture, such as dress and slang, and began moving away from traditional white culture.

Fast foreward to the late 1940's. Jazz has progressed from Big Band and swing to Bop, and there is a massive influx of WWII veterans who felt extremely alienated by the world around them. This group, which became known as The Beats, embraced elements of Bop Jazz, black culture in general, Folk culture*, the Modernist Poetry movement, Buddhism, the drug and gay cultures, and numerous other non traditional sources to find the answers they felt traditional American culture was not providing.

The Beats became the first major counter cultural movement. From there, the hippies adapted Beat culture to fit the needs of a world in the midst of the civil rights movement and Vietnam, Punk Rock responded to the way they felt the hippies failed/sold out/etc, and so on. Despite the differences in each of these movements, all of them can be traced back to the bridging of white and black culture during the first half of the twentieth century.

*There was also a movement of outlaw worship that began in the wake of the civil war, with Frank and Jesse James. This movement spread throughout rural America through the use of folk songs which told "the other side of the story" of a number of outlaw figures. These songs were staples of folk music, and because of this, they ended up following folk music into the world of counterculture. On a tangentially related noted, I argued in an undergraduate research paper that these folk song anti-heroes showed that there is a direct line from Robin Hood (whose stories were brought into America via Howard Pyle's wildly successful collection of stories, released around the time the James brothers were becoming legendary figures, and whose stories were also originally spread by ballad) to modern Counter Culture.