But while beating the very best has often proved well within their grasp, Glasgow have failed to win more than two games in a row across each of their 11 seasons in the Heineken Cup.
And after being drawn alongside reigning champions Leinster, former winners Bath and Top 14 finalists Montpellier in this year's competition, Lineen knows Glasgow will have to buck that trend if they are to reach the knockout stages for the first time since 1998.
"We had three wins in the Heineken Cup last season. It's now about how we get consistency into our game," said Lineen.
"We've always managed to have one or two great results but then fade away. We've got to work out how to get that consistency and keep that momentum up.
"We can stand toe to toe with the big teams, there's no doubt about that. We haven't actually played a bad game so far this season. We've had a couple of loses but they've been very close and the guys have been really consistent in that way. We've had a couple of good away wins against Leinster and the Blues in the league when a lot of the youngsters were playing.
"But it's one thing doing it in the league and another one doing it in the Heineken Cup against Europe's best. This is the big stage now.
"We really need to find that consistency, though, because Scottish teams really haven't done anything in the Heineken Cup. We want to make sure we deserve to be in the tournament, we don't want to just be making up the numbers.
"It's massively important that we challenge for the knockout stages. We want to compete and to win - that's what the game's all about."
Glasgow begin their Pool 3 campaign against Bath at Firhill a week on Sunday and Lineen is acutely aware of how important that particular fixture will be to his team's hopes of progression.
Winning your home fixtures is so often a prerequisite for quarter-final qualification and Lineen doesn't expect this year to be any different.
But taking on one of English rugby's most successful sides, especially now they are led by Lineen's former Scotland coach Sir Ian McGeechan, will be no easy task even on home soil.
"I played under Geech many, many years ago," added former Scotland centre Lineen.
"He's a wily old character and he's been around. He's done everything there is to do in the game and any team that's coached by him is a good team.
"Styles change but I know that he's a very meticulous planner, a very good motivator and an incredibly good rugby coach. He'll have worked out how to play against us but we really just have to concentrate on our game.
"There's healthy respect there, that's for sure. If you look at all the star players they've got down there at Bath, you can have a little bit of fear. But we played Bath four years ago in the Heineken Cup and had two cracking games and Ruaridh Jackson was introduced to Heineken Cup rugby down at The Rec.
"We know how tough Bath first up will be. As we all keep saying, you have to win your home games in the Heineken Cup. That's important against a really top team like Bath."