Wright moved to Deepdale at the end of May after being released by Ipswich but the 34-year-old has not been able to settle in the area without his family and has left North End with immediate effect.
Speaking about the decision, former England goalkeeper Wright told the club's official website: "It might seem somewhat dramatic but I have only been away from my home in East Anglia for a few days and I know already that I simply cannot live that far away from my family.
"I have children settled in school, so moving isn't an option. I think it is only fair to be honest now rather than down the line when the club may not be able to solve the problem that I am creating."
Manager Graham Westley added: "It takes a brave man to be honest in a difficult situation like this. Richard thought he could handle the separations from his family but he has found it too much.
"He is a good man and I wish him well on behalf of all at PNE."
Wright started his career at Ipswich, returning to the club after unsuccessful spells with Arsenal and Everton.