Rhyming verse still dominates Russian poetry. One explanation for this phenomena would be the prevalence of oral vs written poetry distribution mechanisms within Russian/Soviet cultural environment. A long history of political and ideological censorship favors information replication modes that can work around official prohibitions. Rhyming poems are easier to remember or put into a song, which means they can be easily communicated when their publication is either prohibited or limited due to, e.g. a government policy that allows printing of officially approved literature only. It would be important to note here that the official literature, being a creature of conservative literary tastes, tends to favor rhyming verse as well. Therefore we get an environment that for a variety of reasons selects rhythmically "packaged" verse vs its all other varieties.
cf haiku as a predominantly written or even painted verse.