Flanker Wood, 29, is in pre-season training after neck surgery forced him to miss England's tour to Australia.
He has played 129 times since joining in 2010 and replaces scrum-half Dickson, 31, after one year as skipper.
"Lee did a great job and the first thing for me was to congratulate him on that, but to do the job well I need support from guys like Lee," said Wood.
"He's too important for our team not to have that support - he's too good a player and too big a personality around the camp.
"So one of the big things for me was to make sure there was no bitterness, no 'me versus him'.
"When I spoke to Lee I couldn't have been happier with the outcome. I was a bit anxious about doing it, but he was absolutely honest, very dignified, very graceful, pledged his support and felt like the right decision had been made."
Saints had a "good sit down and comprehensive review" after finishing outside the Premiership play-off places last season in fifth.
"It wasn't a case of biting [director of rugby Jim Mallinder's] hand off immediately [when asked about the captaincy]," Wood admitted to BBC Radio Northampton.
"I thought about all the things that come with it because while it's certainly a huge privilege and honour, it's a responsibility and one I felt I had to do justice if I was going to accept the role.
"There was a bit of self-reflection, thinking about is it for me, can I do the role and do everyone proud?
"I obviously came to the conclusion that with the guys I've got around me, like England skipper Dylan Hartley, Lee Dickson and a lot of the senior players in Christian Day, Calum Clark and the like, I'll rely on those guys heavily but I think we can get the job done."
Tom Wood has replaced fellow England international Lee Dickson as captain of Premiership side Northampton Saints.