Wray's successor as Saracens chairman Neil Golding said: "The club is very grateful to Edward for temporarily stepping away from his other commitments to provide significant assistance during a difficult period, and wishes him well in the future."
Griffiths stressed he had never intended his second stint with Saracens to last for long.
"This was always going to be a very short-term appointment, and others are well-placed to drive forward the rebuilding of the club," a statement from Griffiths read.
Saracens said its recruitment process had begun with a view to finding a long-term chief executive.
Griffiths last week denied having a conflict of interest with respect to his activities in cricket. He has held a consultancy role with Middlesex and also refuted acting as an unregistered player agent.
He told London's Evening Standard newspaper last week the England and Wales Cricket Board (ECB) was looking at the claim, saying: "Someone has brought something forward to the ECB.
"They have due process and are required to investigate it. I am very happy to co-operate."