Kean was subjected to familiar calls for his head from the terraces during the defeat to Middlesbrough.
Rovers remain top of the Championship ahead of Saturday's fixtures, but Kean was barracked as his charges turned in a lacklustre display to suffer their first defeat of the season.
Recalled Boro forward Lukas Jutkiewicz opened the scoring in the sixth minute before heading his second after the hour.
Substitute Grant Hanley pulled a goal back amid waves of Rovers pressure in stoppage time but chants of "Kean Out" boomed around Ewood Park at the final whistle.
"I share their frustrations," Kean said when asked about the abuse. "As a staff, as a group, we're frustrated we've lost our unbeaten run.
"I think the fans want to see us get promoted - like the staff, like the owners, like everybody connected to the club.
"We're on a run that's ended tonight, but it's a block of games with a good return.
"I don't want anything else apart from promotion.
The players don't and I've been telling them that since the first day of pre-season.
"I believe we will get promotion, I think we've got a good enough group.
"And would we have taken this amount of points after this amount of games? Yes, because we're on track."
Kean conceded Middlesbrough deserved the points but felt Rovers should have been awarded a penalty when Jordan Rhodes went down under a challenge from Justin Hoyte moments after Hanley reduced the arrears.
"I felt we left it too late, we never really got started," he added.
"Having said that, if the penalty's given - I think it was (a penalty) - then we might have ended up getting something out of nothing.
"Our unbeaten run is gone, which we were so eagerly trying to keep.
"But we're top of the league still, we're averaging two points a game and if we do that to the end of the season then we'll be probably champions."
Middlesbrough manager Tony Mowbray watched his team claim their first points on the road this season and produce the desired response from Tuesday night's heavy defeat at Blackpool.
"I think we were looking for a reaction after Tuesday," he said. "I think Blackpool gave us a football lesson on Tuesday if we're honest and we needed to bounce back.
"I'm really pleased with them and it gives us a platform. We beat Ipswich on Saturday and, six points out of nine, we'd probably have taken that at the start of the week."
On his opposite number's situation, Mowbray feels such hostility might have worked to Rovers' advantage in the season's opening weeks.
"I'm reasonably sure he's using a siege mentality with his group," he said.
"I would assume that they would blank out the negativity from the terraces and try and use it as a strength.
"It would suggest they've been doing that because they were unbeaten and top of the league - unbeaten at home where the environment is probably at its worst for him.
"I wouldn't put too much stock in the fact that the crowd work against the team because I'm pretty sure he's using that."
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