Dinmukhamed (Dimash) Akhmedovich Kunaev was born on January 12, 1912, in the city of Verny (now Almaty). His ancestors were from the Elder Zhuz and practiced the traditional Kazakh occupation of cattle breeding in the Kuigan area along the Kurty and Ili rivers. His grandfather, Zhumabay, completed Muslim school, later worked as a teacher, and made a pilgrimage (hajj) to Mecca.
His father, Minliakhmed (Akhmed) Kunaev, was born in 1886 and was literate in Arabic script. He worked in various roles for a prominent Verny merchant, Iskhak Gabdulvaliev, eventually becoming a clerk. His mother was Zaure Kunaeva. Dinmukhamed grew up in a large family with two brothers and eight sisters, though his younger brother Edige died in infancy and a younger sister, Gulzhanat, passed away at age 11. His siblings pursued various professional paths, with many, like his older sister Amina, moving to Moscow for education. His sisters Amina, Maymuna, Fazilya, Nailya, Gaukhar, Sara, and Rauza, and his brother Askar, all received higher or specialized education and made careers in fields ranging from medicine and engineering to science and cultural work.