He became a music reporter for the Czech Radio studio in Brno while studying musicology at Masaryk University under Bohumír Štědroň and Jan Racek.
For the years 1953–1956, Podéšť replaced Radim Drejsl (1923–1953) as director of the Vít Nejedlý Army Artistic Ensemble (Armádní umělecký soubor Víta Nejedlého) in Prague, then during 1958–1961, he worked as an editor of music broadcasts for Czechoslovak Television.
1, the piano fantasies Písně smutné paní, the Woodwind Quintet and Cello Sonata), but later found inspiration in folklore and in the work of the Moravian composer Leoš Janáček (for example the symphonic poem Raymonda Dienová, the orchestral rhapsody Advent and Maminka for children's chorus and orchestra).
Following the suicide of Radim Drejsl he became one of the most important Czech composers of the politically engaged songs called "budovatelské písně".
Podéšť is the author of several professional journal studies such as Za odkazem V. Nejedlého (Vít Nejedlý Legacy; Hudební rozhledy, Vol.