Shiels was sacked by the Rugby Park club after a succession of run-ins with the Scottish Premier League and the team's poor form at the end of the season, which resulted in a ninth-place finish.
But the 57-year-old believes everyone at the club should now focus on the future, and back his replacement.
"Obviously there's parts of me that makes me feel well 'thank you very much lads' that was good to have that support and that support is still there," Shiels told Sky Sports News.
"The other side of it is I feel as if Kilmarnock should move on. Kenny Shiels is not there anymore, and as much as I appreciate the supporters doing that, I would like them to try and get behind the new manager, the chairman.
"I'd go back to Kilmarnock tomorrow, if everybody were mutually agreed to that, but that's not really fair because I want to move on now to somewhere else.
Shiels insists he has no misgivings about his time with Killie, as he was simply trying to do his best for the club.
"I don't give two hoots what people say about me, I really don't. As long as they appreciate and respect the work that I do," Shiels added.
"The only regrets I have is not being able to see out the potential that's there at my club."