Atleti have reached at least the quarter-finals of the competition in four of the past five seasons while making the final in both 2013-14 and 2015-16, losing to local rivals Real Madrid on those two occasions.
There will be no repeat in the Europa League however, as Atletico advanced past the group stage this season and face Italian giants Juventus on Wednesday in the first leg of their mouth-watering last-16 tie.
Juventus also can boast of two trips to the final in a recent three-year span, going in 2014-15 and 2016-17, losing to Barcelona and Real Madrid respectively.
And lack of motivation is unlikely to be an issue against the runaway Serie A leaders, as Griezmann believes the tantalising potential of a home final could give Diego Simeone's side a vital edge.
"It gives us and the fans even more motivation," he told UEFA.com.
"There's always a great atmosphere and that's what we need, that's what we want as players. A home final would be a celebration for everyone associated with Atletico.
"The path there will be very difficult, but we will try our best. We will do everything in our power to get there, and then we'll see."
Atleti have largely kept intact the side that won last season's Europa League and Griezmann believes that continuity – and experience of big occasions – will stand them in good stead as they enter the Champions League knockout stages.
"We have players that have already played in Champions League, Europa League and World Cup finals," he added.
"We're already gaining in confidence in important games and moments and I think it's at those times that we really raise our game and that helps us play better.
"We have new faces in the team, but I think that the fundamental base remains the same and that is what will help us to go far."