The Harlequins centre will lead Wales in June Tests against Tonga and Samoa while the Lions take on New Zealand.
Roberts, 30, played for the Lions in 2009 and 2013 but was not selected this time by Warren Gatland.
"I was gutted for about a day or two and then you refocus and you go again and you use it as motivation," he said.
"You respond to it in one of two ways. You mope around and continue to mope around, or you dust yourself off and you go again.
"I've played some good rugby in the last few weeks for my club at Harlequins and I'm thoroughly looking forward to this tour.
"And I suppose being captain gives me extra incentive."
Roberts has started 83 of his 91 Tests for Wales but only one since they lost heavily to Australia in November.
"I was disappointed with the autumn series, in particular against Australia," said Roberts.
Roberts acknowledges he "deserved" to be dropped for his performance against the Wallabies.
"It was a bad day at the office and I paid the price for it," he said.
"Hopefully that's given me a kick up the backside to go again, and in the Six Nations it was disappointing not to get a shot at starting."
With 12 Wales players in the Lions squad, Roberts will lead a squad including 13 uncapped players.
It will be his first experience of captaining a side since he led Cardiff Schools Under-15s.
Wales' first Test is part of an Auckland double-header on Friday, 16 June, with their game against Tonga preceding New Zealand's match against Samoa.
A week later, Wales will be in Apia to take on Samoa.
Roberts said: "For me, the fundamental thing is having the time of your life and being accountable.
"I want the young lads to have the time of their lives playing for their country.
"It's a hugely privileged position we're in to play rugby for Wales. It's every boy's dream, really, and I want them to have the most fun possible.
"I truly believe you do your best work when you truly enjoy what you're doing."
Tonga have not beaten Wales in seven attempts, but Samoa have won four of the teams' nine matches.
Roberts said: "The biggest challenge for this group - and we weren't good at it in the Six Nations - is performing away from home.
"We know the comforts of the Principality Stadium, we know the motivation that gives Welsh players. Unfortunately, in the Six Nations, we didn't perform in Scotland and France. That is going to be the challenge.
"To embrace the occasion and pressure is what we want from the players - to embrace the environment and not be afraid of it."
Jamie Roberts intends to use his disappointment at being overlooked by the British and Irish Lions as motivation when he captains Wales.