Bjrken was part of Harry Waagens Orchestra, a cental band on the Trondheim Jazz scene in 1949-53. He later had his own orchestras in Steinkjer (1956-57) and Trondheim (1957-). In the lineups was among others Karl Holst (clarinet and saxophone, 1957-), Kjell Johansen (drums, 1957-) and Bjrn Alterhaug (bass, 1966-). I. 1990 the orchestra consisted of Oddmund Finnseth, Ove Bjrken (guitar), Per Olaf Green and Rolf Skogstad. Av utenlandske samarbeid kan nevnes Bengt Hallberg, Benny Bailey, Mads Vinding.
In the 1980s he played in Egil Kapstad/Rowland Greenberg Quartet, and since 1990 he has led own Asmund Bjrken Swing Sextet, which inter alia Ove Bjrken, Bjrn Alterhaug and Bjrn Krokfoss are included, as well as Erling Aksdal (piano).
He released the autobiography Spellemann, p gammel rutine og stgg mistanke in 2003, and a portrait of him vas put up at the Norwegian television channel NRK in 2003, called Asmund i Himmelriket, by Andreas Lunnan.