After leaving school he joined the Ulster Bank in Granard, Co Longford, and worked in banking for the next five years. However, he remained set on becoming an actor and when he received a posting to Dublin he soon made a mark on the city's amateur scene appearing with the Rathmines & Rathgar Operatic Society, and the Dublin Shakespeare Society. His employers were not impressed by his extra curricular activities and in 1953 he was posted to the remote town of Killeshandra in County Cavan. McKenna refused to go and resigned his position.
McKenna made his stage debut at the Pike Theatre in Dublin in 1953 as John Buchanan in Tennessee Williams' Summer and Smoke. He played a season at the Gaiety Theatre with Anew McMaster's Shakespearean company, and was a member of The Gas Theatre Company directed by Godfrey Quigley.