Referee Peter Walton missed the incident at St Andrew's on Saturday but confirmed to the Football Association that he would have sent Bowyer off had he seen it.
As a result, the FA charged Bowyer with violent conduct and the club have confirmed he 34-year-old has accepted the charge.
Blues boss Alex McLeish said on the club's official website: "We accept the decision and will now have to move forward into the next three games without the services of Bowyer.
"We have got a good discipline record at this club and stress to the players every week how important discipline is."
Bowyer will miss Tuesday's Premier League game against Blackpool, the FA Cup clash with Millwall and the first leg of the Carling Cup semi-final against West Ham United.