The Australian international scored a brilliant goal in the 2-1 loss to the Whites on the weekend in his return to Elland Road following his departure from the club at the end of last season.
Following the match, Kilkenny hit out at Ken Bates after comments the Leeds owner made on radio on Saturday morning and defended his decision to celebrate his goal against his old club.
"I didn't really want to celebrate in front of the fans," Kilkenny told the Yorkshire Post. "But I needed to make a point to the chairman and his chief executive because everyone here has said hello to me except for the chairman and the chief executive, which is disappointing.
"I have seen a few things said by the chairman over the last couple of days. He said that I left Leeds to go to a smaller club and that was very disrespectful to both me and to Bristol City.
"That is really, really disappointing. But that's him, what can you do?"