Turtle died on his way to the vet.
"A witness confronted the guy but he shrugged his shoulders and took off," the cat owner said.
The couple say they are not angry with the dog, but the owner.
"He didn't have control over his dog and that's not good enough, and he left immediately.
"At the least he could have stuck around and explained what happened and offered an apology.
"The cat was a cherished family pet, a real character, who we found on the road as a kitten and took him in. Our two young children are devastated and understandably have suffered a huge emotional impact.
"There's nothing more we can do without knowing where the guy lives so we can lay a complaint.
"What or who will be the dog's next victim if it's allowed to roam unleashed - another dog, or a young child?"
Rotorua District Council spokesman Rex Moore said the council understood this had been distressing for the owners of the cat.
Animal control officers had investigated but had unfortunately been unable to identify the dog responsible at this stage, he said.