Rovers remain top of the npower Championship ahead of the rest of the weekend's fixtures, but Kean was barracked as his charges turned in a lacklustre display to suffer their first defeat of the season.
"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."
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 think the fans want to see us get promoted - like the staff, like the owners, like everybody connected to the club," Kean added.
"We're on a run that's ended tonight, but it's a block of games with a good return."
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."
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'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," he said.