A member of an amateur Gaelic theatre group, McKenna made her professional stage debut in 1940 at the Gaelic repertory theatre An Taibhdhearc in Galway, Ire. In 1951 she starred in her own Gaelic translation of Saint Joan, a role she repeated in English in London (1955) and in the United States (1956). She appeared with the Abbey players in Dublin in both Gaelic- and English-language plays (1943–46), then made her London debut in Paul Vincent Carroll's White Steed (1947). McKenna first achieved international fame for her role as Pegeen Mike in 1951 at the Edinburgh Theatre Festival. She also performed in the motion picture version of the play (1962).

In 1961 she appeared as Joan Dark in Bertolt Brecht's Saint Joan of the Stockyards, repeating the role in 1964. She was also memorable as Juno in Sean O'Casey's Juno and the Paycock and as Josie in Eugene O'Neill's Moon for the Misbegotten. Her one-woman readings from Irish authors were also popular.