Born and raised in Hamburg, Griem entered featured with a small role in "East Zone, West Zone" (1958) but it was over a decade before he was "discovered" by Visconti. Utilizing the actor's handsome features and diffident air, Visconti cast him as the aristocratic Aschenbach, the bisexual with ties to the SS. Fosse played on the same qualities when he tapped Griem to portray Max, the embodiment of corruption in Weimar Germany. The actor turned in another strong performance under Visconti's eye as the army soldier with an unwavering devotion to Helmut Berger's insane monarch "Ludwig".
From the mid-70s on, Griem often appeared in German films with political themes. In "Children of Rage" (1974), he was miscast as a Jewish doctor caught up in the tensions between Israelis and Palestinians, with Cyril Cusack as his father and Simon Ward as his brother. Griem was typically cast as the outsider who becomes enmeshed in matters with political consequence in such features as "Deutschland im Herbst/Germany in Autumn" (1978), "Mannen i skuggan/Black Sun" (1978) and "Kaltgestellt/Put on Ice" (1980). He delivered a strong supporting turn as a vicious ship's steward imperiously treating the Jewish passengers in the bloated all-star "Voyage of the Damned" (1976). On paper Griem seemed perfectly suited to play an architect wrongly imprisoned over a faultily designed school in "Die Glaesserne Zelle/The Glass Cell" (1978), but his performance was somewhat slack and monotone. Even when his character was driven to murder, the actor seemed curiously unimpassioned. He fared better as the doomed colleague of Richard Burton in "Breakthrough" (1979), an angel in Rainer Werner Fassbinder's epic "Berlin Alexanderplatz" (1980) and in the title role of "Faust" (1989). More recently, he had a small role as an art dealer in the German comedy "Verlassen Sie Bitten Ihren Mann!/Please, Leave Your Husband" (1993), his last screen appearance to date.
By the mid-80s, Griem began to crack the US TV market with appearances in the NBC miniseries "Peter the Great" (1986) and the CBS TV-movie "The Plot to Kill Hitler" (1990).