"We believe their voices should be heard throughout our country," spokesman Aaron Harison said in a statement.
Those featured in the video, some looking over the former site of the World Trade Center from a nearby building, recount their traumatic stories of that day. Their opposition to the cultural center isn't mentioned until halfway through the ad, and the site itself is never shown.
"We promised 'Never Forget.' Is that just meaningless words on a bumper sticker faded by the sun? Give me a break," said firefighter and 9/11 survivor Robert Reeg.
Lee Hanson, whose son, daughter-in-law and granddaughter were aboard the second plane which hit the South Tower on Sept. 11, indicated he understood that Muslims had the right to build the center on private property.
"We understand they can build where they want to build," Hanson said. "All we ask them to do is to use some good sense and think about what they're doing."
Cheney and Kristol are both board members of the group, as is Debra Burlingame, the sister of Charles F. "Chic" Burlingame, III, the pilot of the plane which crashed into the Pentagon on Sept. 11. Burlingame is also the head of 9/11 Families for a Safe and Strong America.
The video is embedded below.
(Ed. note: This post has been updated.)