According to Victor Sonkin, the generation that grew up in the Soviet Union in the 1970s and '80s listed Kurt Vonnegut, who died two weeks ago among their favorite authors:
One was Vonnegut's life story and his aversion to war. Enlisted in the U.S. Army during World War II, he was captured by the Germans and was one of a handful of American POWs who survived the Allied bombing of Dresden in 1945; he was later freed by Soviet troops. This experience formed the core of his novel "Slaughterhouse-Five." Throughout the postwar era, both official Soviet propaganda and popular feelings were strongly antiwar (even the infamous Afghan campaign was never heralded in belligerent terms), so Vonnegut was in tune with the nation's mood.
Another reason was that he wrote science fiction, one of the few ways for writers to address important issues that would have been censored in other genres. "Cat's Cradle" was about scientists' (and society's) responsibility; the seminal short story "Harrison Bergeron" showed how egalitarianism could turn into tyranny. Such issues, taboo in everyday Soviet writing, could be smuggled in through science fiction and enjoyed considerable success.
Third was Vonnegut's style. This usually gets lost in translation, but Vonnegut was lucky to have Rita Rait-Kovalyova as his translator. In one of Sergei Dovlatov's satirical sketches, someone asks him who has the best prose style in Russian. He says, "Rita Rait," and the reaction is, "You mean Vonnegut in Russian is better than Fedin? How awful." (Konstantin Fedin was an official Soviet writer and bureaucrat.)
Finally, it was just sheer chance. No book by a living foreign author, especially an American, could appear in the Soviet Union without the blessing of the Party. Vonnegut was, in a sense, authorized. This explains the extent of his popularity, which other authors of a comparable caliber, such as Saul Bellow or Joseph Heller, did not achieve here.