Kurtz was married three times. He was married to the American flautist Elaine Shaffer until her death in 1973. After her death, Kurtz married Mary Lynch, who survived him. He died in London, aged 94.
His recorded repertoire included works by among others Dmitri Shostakovich (early recordings, though not premieres, of several of the symphonies and the Age of Gold ballet suite), Ernest Bloch (one of whose last works, Two Last Poems (Maybe...) was dedicated to Elaine Shaffer), Heitor Villa-Lobos (his Uirapuru). Many of these recordings made in the 1940s and 1950s with the Philharmonia in London. He recorded primarily for Columbia Records and EMI.