They say the movie is crude anti-Soviet propaganda that distorts history and have called for it to be banned from Russian screens.
Indiana Jones and the Kingdom of the Crystal Skull stars Harrison Ford as an archeologist in 1957 competing with an evil KGB agent, played by Cate Blanchett, to find a skull endowed with mystic powers.
"What galls is how together with America we defeated Hitler, and how we sympathised when Bin Laden hit them. But they go ahead and scare kids with Communists. These people have no shame," said Viktor Perov, a Communist Party member in Russia's second city of St Petersburg.
The comments were made at a local Communist party meeting and posted on its internet site www.kplo.ru.
The film, the fourth in the hugely successful Indiana Jones series, went on release in Russian cinemas yesterday. Russian media said it was being shown on 808 screens, the widest ever release for a Hollywood movie.
In past episodes Indiana Jones has escaped from Nazi soldiers, an Egyptian snake pit, a Bedouin swordsman and a child-enslaving Indian demigod.
"Harrison Ford and Cate Blanchett (are) second-rate actors, serving as the running dogs of the CIA. We need to deprive these people of the right of entering the country," said another party member, Andrei Gindos.
Though the ranks of the once all-powerful Communist Party have dwindled since Soviet times, its members see themselves as the defenders of the achievements of the old Soviet Union.
Other communists said the generation born after the 1991 collapse of the Soviet Union were being fed revisionist, Hollywood history. They advocated banning the Indiana Jones outright to prevent "ideological sabotage."
"Our movie-goers are teenagers who are completely unaware of what happened in 1957," St Peterburg Communist Party chief Sergei Malinkovich told Reuters.
"They will go to the cinema and will be sure that in 1957 we made trouble for the United States and almost started a nuclear war."
"It's rubbish ... In 1957 the communists did not run with crystal skulls throughout the US. Why should we agree to that sort of lie and let the West trick our youth?"
Vladimir Mukhin, another member of the local Communist Party, said in comments posted on the internet site that he would ask Russia's Culture Ministry to ban the film for its "anti-Soviet propaganda".
The Indiana Jones film is not the first Hollywood production to offend Russian sensibilities.
In 1998 the Russian parliament demanded the government explain why the Hollywood film Armageddon - which depicted a dilapidated Russian space station that blows apart because of a leaky pipe - was allowed onto Russian cinema screens.
A government official at the time said the film, starring Bruce Willis as the leader of a team of astronauts sent to deflect an asteroid on a collision course with Earth, "mocked the achievements of Soviet and Russian technology".