The 31-year-old has signed a one-year deal with an option for a further season at Kuban, who finished fifth in the Russian Premier League last season to qualify for the Europa League.
The journeyman striker, for whom Kuban will be his fifth club since leaving Marseille in 2009, hopes his return to the European stage will catch the eye of France boss Didier Deschamps.
"I love my country, I love the sensations that feel when I put on a tricolour shirt," said Cisse, who last turned out for Les Bleus in 2011.
"I really want to give all my power to once again return to the camp of the national team.
"I understand that I need to have a very good season for my new club.
"I think if Kuban achieve all their goals in the coming season, and I score regularly, anything is possible."
Cisse had a year left on his contract at QPR, but reached an agreement on a mutual termination earlier this week.
He joined Rangers from Lazio for a reported £4million in 2012 and went on to score 10 goals in 29 appearances, but spent the second half of last season on loan at Qatari outfit Al-Gharafa.
With QPR boss Harry Redknapp required to reduce the wage bill following the club's relegation from the Barclays Premier League, Cisse's high wages were always likely to prove troublesome, and the player admitted he had been anticipating his exit.
Cisse said the move to Kuban was therefore not a snap decision, with the club first making contact several weeks ago.
"Overall, it was about three weeks," he said. "I had to pause to think about it calmly because it was clear that I was leaving QPR.
"Therefore the decision to move to Kuban was quite a balanced step."