The American, from Utah, was on Westminster Bridge with his wife Melissa, 46, when they were mowed down by Khalid Masood's auto. Meanwhile, a picture of a Muslim woman, donning a Hijab and walking by an injured person on the Westminster bridge after the attack, went viral on social media as many "slammed her for lack of concern".
"We know that Kurt wouldn't bear ill feelings towards anyone and we can draw strength as a family from that", Payne said.
Deputy Assistant Commissioner Neil Basu said it may never be possible to fully determine the motives of attacker Khalid Masood, who was shot dead on Wednesday after running over pedestrians with an SUV on London's Westminster Bridge and fatally stabbing a policeman guarding Parliament.
In the first public statement by a victim's family since the attack, Payne thanked the emergency services and members of the public for "the outpouring of love and generosity".
Sarah Waseem (57) from Surrey, said: "When an attack happens in London, it is an attack on me".
Taking questions from reporters, the family said they were trying not to concern themselves with the attacker's motivations and the suggestion he carried out the attack in the name of religion.
Mr Cochran's brother-in-law, Clint Payne, told a press conference at New Scotland Yard: 'The most hard part of all of this is that Kurt is no longer with us, and we miss him terribly.
US President Donald Trump announced Cochran's death last Thursday, calling him "a great American" in a Twitter post.
Echoing her mother, Jennifer Burton described her sister as a feisty and strong-willed fighter.
Mrs Cochran, 46, is still being treated in hospital for her injuries.
They had been due to fly back to the U.S. the day after the attack.
They thanked people who have supported them and said, "We care about him being remembered for his selfless bravery and loving nature".
"I do not condone his actions nor support the beliefs he held that led to him committing this atrocity", she added.
"Everyone has been wonderful", he said. You did your best and we are just grateful he was not alone. "That was just who he was, he was happy to share in what we had created, and he was so excited that we were able to accomplish something with our band". "That's really going to be a tough one for our little family members".
Cochran's father-in-law said people had been "amazing". "We're actually kind of excited to get back home and give everybody a hug".