The striker's miss against Motherwell was his fourth of the campaign.
The Bhoys' top goalscorer squandered the opportunity to put Ronny Deila's men ahead during their game with Motherwell on Saturday after putting his effort wide.
And Kennedy has seemingly confirmed that the club will actively seek to give someone else the responsibility for the next time a chance from 12 yards arises.
"At the start of the season we have a list in order of penalty takers and it was Leigh and Kris Commons who were first and second," he told the Evening Times.
"So we will have a look at this week and see how one or two others feel about putting their hands up."
With a Scottish Cup semi-final ahead, it has arguably become even more important that the Scottish Premiership leaders find a taker who has the confidence to cope with the cauldron of pressure if they get the chance at Hampden Park.
And the decision to give someone else the role may actually prove to have a positive impact on Griffiths. While his ability means that he will always be favourite to score from the spot, the fact that the 25-year-old has missed so many in a short space of time may see him concentrating too much - and perhaps worrying - on the prospect of potentially taking another penalty rather than focusing on how he can inspire Celtic to the final.