During the Great Patriotic War, his family was evacuated to the village Pyschug [ru] in Kostroma Oblast, where Kunyaev completed his first four grades of primary school.

While at university, Kunyaev started writing poetry, and his first collection of poems Earth Explorers was published in 1960.

Between 1957 and 1960, he worked for the newspaper Zavety Lenina (Tayshet, Irkutsk Oblast), before serving as head of the poetry department of literary magazine Znamya from 1960 to 1963.

In 1967, having considered that he was in a creative crisis, he left for five years to work in geological parties in the Pamirs, Tien Shan, and the Hissar Range.

Kunyaev has written around 20 books including Eternal Companion, Scroll, Manuscript, Deep Day, and Favorites, and has translated poetry from Ukrainian, Georgian, Abkhazian, Kirghiz, Buryat, and Lithuanian.