The son of former Bradford chairman, Geoffrey, he was an important part of the consortium that saved Leeds from going out of business last March.
"His decision has come as a surprise," Leeds chairman Gerald Krasner told the club's official website. "He has done a wonderful job with the players and without him we would have had big, big problems."
Krasner said the club's chiefs would meet soon to discuss their future structure.
"Hopefully David will be around to help us with the changeover," added Krasner. "The board wish him well and we are very sorry to lose him."
Rumour has it that Richmond was getting concerned about the supporters' reaction to the sale of players and the proposed sale of the ground.