Kurtz was married three times. He divorced his first wife, Katherine, to marry flutist Elaine Shaffer; Shaffer had been the first-chair flute of the Houston Symphony (under Kurtz) until she resigned in 1953 to pursue a successful solo career. The two remained married until her death, of lung cancer at the age of 47, 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.