All they did was ask.
The guide on our tour of the Baltics last year, a Lithuanian woman, said all the Lithuanian government did was ask to leave.
It was written into the Soviet constitution ( I'm not sure this is the correct word, but it conveys the sense) that any country could ask to leave. It hadn't occurred to any country to ask until hers did it.
The above is a simplistic explanation of how Lithuania regained its independence, but it is how it started.