Wood, 27, played 74 games in his three years with the club and scored 11 tries. Johnson, 22, played 35 games in total for the club.
Giants managing director Richard Thewlis outlined the reasons for letting the players go.
“In the case of Josh we offered him a very good contract and one that would have significantly increased his salary last year, as well as going forward,” he told his club’s official website.
“So it was particularly disappointing to everyone here that he chose to move on.
“We were talking to him since the beginning of the season but in reality I don’t think we ever got close to the valuation which he was seeking and we wish him well for the future.
“We are fortunate in the prop area to have younger players Tyler Dickinson and Matthew English who now get closer to a top 25 squad place after Josh’s departure.
“In respect of Kyle he has an excellent opportunity elsewhere which he has taken and we wish him well.
“He suffered a number of injuries during his time with but always battled back and was a popular member of the squad.