The victims, who identify themselves as Dr Abdulmasih Aniya, Ashur Abraham and Bassam Michael, were among a group of around 253 Christians IS has kidnapped from north-eastern Syria in February.
In the video the men are dressed in orange jumpsuits, similar to the murders of 21 Coptic Christians in February, before militants shoot them dead.
The Guardian has said Islamic State still has approximately 202 of the 253 Christian captives from February.
In the video, the extremists threaten to kill the remaining 202 unless a £65,000 ransom is paid for each person.
An Assyrian activist network which withheld its name for security reasons told the newspaper in a statement: "We condemn this latest act of barbarism in the strongest possible terms.
"The systematic ethno-religious cleansing of Assyrian/Syriac/Chaldeans continues. They are helpless. They are children. They are women. They are somebody's father and brother.
"We plea and beg of the international community to intervene immediately. We have been driven out of our ancestral lands. We have been killed and crucified. The international community must act now to save lives of others kidnapped."
The news comes as a former British diplomat to Syria, Peter Ford, said Christians in the country would be massacred if President Bashar al-Assad is toppled.
According to the Church Times, Peter Ford also said Prime Minister David Cameron would have "blood on his hands" if he helped oust his Syrian counterpart.
Mr Ford also said there's no such thing as moderate opposition groups in Syria - only pro-government forces and Islamist terrorists.
Syrian President Bashar al-Assad is accused of killing almost 180,000 of his own civilians since 2011, including the use of chemical weapons against them.
Organisations such as the United Nations, Human Rights Watch and Amnesty International have all accused Mr Assad of war crimes and crimes against humanity.
Russia, a longstanding ally of Syria's, has recently started using airstrikes to destroy what they call "well known" terrorist groups operating in the country.
The United States and other nations have accused Russia of purposely targeting moderate groups opposing what they say is Mr Assad's regime.
Steven Carter from the Christian charity Middle East Concern is calling on believers to pray for Christians who remain in danger.
Speaking on Premier's News Hour, he said:"We are certainly wanting to pray for protection, for safety. We are wanting to pray that they would be guided to safe places.
"Most have left those areas that are controlled by Islamic State but many are taking refuge in other parts of Syria or neighbouring countries."
Listen to Steven Carter speaking to Premier's Hannah Tooley on the News Hour:
