Given received a medal following City's Wembley triumph over Stoke in May, but he could take little personal satisfaction from the achievement having played no part in the game.
Having acted as back-up to Joe Hart last season, Given made the move to Villa earlier in the summer and is now looking forward to playing on a weekly basis.
The Irishman values regular game time as the most important thing but would still love to win some silverware with his new club.
Reflecting on his FA Cup victory with City, Given told the Daily Mirror: "I was happy for the team and the fans, but I didn't feel I had contributed because I didn't do anything.
"Maybe I didn't deserve a medal. It's in the house somewhere.
"If I could win something with Villa, it would mean more to me. That has to be our target this season."
Given is grateful to McLeish for taking him to Villa after a difficult year with City.
"It's great to feel wanted again," said Given. "I thank McLeish for bringing me to the club. He had other choices and it's nice to be part of the group and knowing that you're going to be playing at the weekend.
"It's important to feel wanted. I just want to play football and I've shown all through my career that if I wasn't playing, then I wasn't a happy person. It doesn't matter if it's the biggest club in the world or the smallest, I just want to play.
"The word frustrated has been overused by me. Every time you listened to me last season, it was used. But it was very frustrating training all week, knowing you're not going to play.
"It's all water under the bridge now though and I'm just looking forward to playing regularly in the Premier League again."
Speaking about McLeish, Given added "I think he's a great man-manager. He gets the best out of the players.
"He listens to the lads and takes on board their opinions. It's not his way or the highway, he does take everything on board. We're in it together as a group and that's important.
"He's a player's manager. One of his greatest attributes is the way he speaks to people and treats people, and as a player you respond positively to that."