Woodward is also director of elite performance for the British Olympic Association and is expected to operate in a largely advisory role at Welford Road.
The 2003 World Cup winner said: "I consider it an honour to be appointed to the board of directors at Leicester Tigers, a club that has set benchmarks for others to follow throughout its history."
Tigers chairman Peter Tom said: "John Allen and Bob Beason have been associated with the club for more than 40 years, and David Matthews for more than 50 years.
"They were all players for Leicester and members of the general committee before becoming members of the board.
"They have made an invaluable contribution in both the amateur era and since the game went professional and Leicester Football Club became a public company in 1997.
"Their retirement provides us with an opportunity to revitalise and strengthen the board to meet the new challenges that lie ahead, not least the major plans we have for the redevelopment of Welford Road.
"I am delighted to be able to welcome Sir Clive, Rory and Terry to the board and I am certain that their individual and collective contribution will prove vital to helping us achieve our vision of making Tigers the best club in world rugby."
Former Tiger Underwood, now a management consultant, said: "Leicester Tigers is a special club right at the top of the game both in England and in Europe.
"I look forward to being able to play my part in the club's continued successes on and off the pitch."