While awaiting a Colonial Service posting to the colony of Nigeria, Kennedy embarked on a career in songwriting. His first success came in 1930 with "The Barmaid's Song", sung by Gracie Fields. Fellow lyricist Harry Castling, introduced him to Bert Feldman, a music publisher based in London's "Tin Pan Alley", for whom Kennedy started to work. In the early 1930s he wrote a number of successful songs, including "Oh, Donna Clara" (1930), "My Song Goes Round the World" (1931), and "The Teddy Bears' Picnic" (1933), in which Kennedy provided new lyrics to John Walter Bratton's tune from 1907.
In 1934, Feldman turned down Kennedy's song "Isle of Capri", but it became a major hit for a new publisher, Peter Maurice. Kennedy wrote several more successful songs for Maurice, including "Red Sails in the Sunset" (1935), inspired by beautiful summer evenings in Portstewart, Northern Ireland; "Harbor Lights" (1937); and "South of the Border" (1939), inspired by a holiday picture postcard he received from Tijuana, Mexico, and written with composer Michael Carr. Kennedy and Carr also collaborated on several West End shows in the 1930s, including London Rhapsody
(1937). "My Prayer", with original music by Georges Boulanger, had English lyrics penned by Kennedy in 1939. It was originally written by Boulanger with the title "Avant de Mourir" in 1926.
During the early stages of the Second World War, while serving in the British Army's Royal Artillery, where he rose to the rank of Captain, he wrote the wartime hit, "We're Going to Hang out the Washing on the Siegfried Line". His hits also included "Cokey Cokey" (1945), and the English lyrics to "Lili Marlene". After the end of the war, his songs included "Apple Blossom Wedding" (1947), "Istanbul (Not Constantinople)" (1953), and "Love Is Like a Violin" (1960). In the 1960s Kennedy wrote the song "The Banks of the Erne'", for recording by his friend from the war years, Theo Hyde, also known as Ray Warren.
Kennedy was a patron of the Castlebar International Song Contest from 1973 until his death in 1984 and his association with the event added great prestige to the contest.