She married Joseph Kennedy on May 15, 1953, a month after graduating from Ohio State, and the couple had two children, Joseph Jr. and Adam. They divorced in 1966. Her first play to be produced was "Funnyhouse of a Negro", a one-act play she wrote in 1960, the year she visited Ghana for a few months with her husband on his grant from the African Research Foundation. The play draws on Kennedy's African and European heritage as she explores a "black woman's psyche, riven by personal and inherited psychosis, at the root of which is the ambiguously double failure of both rapacious white society and its burdened yet also distorted victims." "A Rat's Mass" was produced at La MaMa Experimental Theatre Club in Manhattan's East Village twice in 1969 and once in 1971. In 1976, La MaMa's Annex performed the show with music by Cecil Taylor. "Sun: A Poem for Malcolm X Inspired By His Death" and "A Beast Story" were both produced at La MaMa in 1974. As of 2018, Kennedy has written thirteen published and five unpublished plays, several autobiographies, a novella, and a short story. Kennedy used the alias Adrienne Cornell for the short story "Because of the King of France", published in "" in 1963. Much of Kennedy's work is based on her lived experience. Kennedy was a founding member of the Women's Theatre Council in 1971, a member of the board of directors of PEN in 1976–77, and an International Theatre Institute representative in Budapest in 1978. Suzanne Alexander is a recurring character in several of Kennedy's plays. "She Talks to Beethoven", "The Ohio State Murders", "The Film Club", and "The Dramatic Circle" are collectively known as the Alexander Plays, and were published together under that title in 1992. A letter written from Suzanne Alexander's perspective, "Letter to My Students on My Sixty-First Birthday by Suzanne Alexander," was also published in 1992. The Alexander plays are less overtly surreal than many of Kennedy's earlier works, but still avoid linear narrative. In the foreword to the printed collection of plays, Alisa Solomon, professor in the Graduate School of Journalism at Columbia University, wrote: "the action of these plays is made up not of the events of Suzanne's life but of the process of turning memory into meaning." She won several awards for her plays, including two "Village Voice" Obie Awards. Her Obie Awards were for "Distinguished Play" in 1964 for "Funnyhouse of a Negro" and "Best New American Play" in 1996 for "June and Jean in Concert" and "Sleep Deprivation Chamber".