edit: not in the 60's, but might be interesting.
It was very difficult for any movement that was counter to the soviet state to develop, as most public activities and public spaces were fully controlled by the state. However, some historians might argue that the "shestidesyatniki" (or "people of the [nineteen] sixties") could be considered somewhat analogous. http://boundary2.dukejournals.org/content/36/1/95.abstract This historian, for example, claims that there were shared connections and motivations present between this movement and counterculutre movements in the west at the same time. These shestidesyatniki were intellectuals, in the sciences and culture, hoping there would be a more open and free exchange of ideas after Stalin died. They worked towards this, challenging the state in fairly gentle ways (publishing somewhat critical articles, etc). They hoped a more open relationship with the west might be possible. Instead, after an uprising in Czechoslovakia by similar communists seeking change in 1968 was put down forcibly by a Soviet military invasion, they realized this was not going to happen. (see "Zhivago's Children", by Vladislav Zubok.) Youth movements that were more openly challenging to the government came later, in the late 1970s and 1980s, particularly after the policy of "glasnost" (openness) brought in by Gorbachov, the final leader of the USSR. As a result of this policy, from 1985-1990, everything opened up more in the Soviet Union--people could protest, speak and publish more freely--and counterculture movements became more sustainable.