Armed Austrian police arrested Aslan Gagiev in January 2015 at a Vienna train station. His gang is accused by Russian authorities of at least six murders.
Those killed include the deputy vice-president of the Russian republic of North Ossetia, the deputy public prosecutor, a mayor and the chief of police.
After several different rulings, Austria's highest court last year put his extradition to Russia on ice because of concerns about how he would be treated there.
This was in part because Russia's constitutional court ruled in July that Russian laws take precedence over judgements by the European Court of Human Rights (ECHR).
Gagiev, 45, says he is the victim of a case of mistaken identity and says that Russia's attempt to have him extradited is politically motivated, the Austria Press Agency reported.
His bond was set at €100,000 ($110,000) and Gagiev undertook not to leave the country. Bulgaria also wants him prosecuted for allegedly using a fake Bulgarian passport.