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The night after Vladimir Putin was inaugurated for his third term as president, a trio of brothers sat down in a smoky piano bar to discuss the politician’s return. Co-hosts of the Web TV show Dzyadko 3, Filipp, Timofei, and Tikhon Dzyadko have become three of the most recognizable faces of Moscow’s liberal opposition. Each week, the brothers air their show—which has become a de facto anti-Putin brainstorming session—on Dozhd, an independent channel broadcast mostly on the Internet. During the protests in Moscow last winter, Dzyadko 3 was directly responsible for getting thousands of people out onto the streets (the brothers refer to the unrest as “the movement of unpleased individuals”). Now, as activists rushed to...