With the international attention given to the trial and conviction of members of the Russian punk band Pussy Riot on charges of "hooliganism," many have wondered online whether Pussy Riot is a translation of a Russian name. But no: the band consistently uses Pussy Riot (in Latin characters) on its official LiveJournal blog, even though most of the text is in Russian (in Cyrillic characters). This isn't too surprising among punk/alt-rock bands worldwide. Whether it's the Japanese noise rockers Boredoms or Russian ska-punks Distemper, musicians very often use English in Latin script for the names of their bands (and titles of albums and songs), even when their lyrics are in their native language. But how have Russian sources identified Pussy Riot?
Most media outlets have followed the band's lead and have used Pussy Riot even in Cyrillic text (e.g., here, here, and here). Others have transliterated the name of the band into Cyrillic characters, as Пусси Райот (e.g, here, here, and here). Cyrillicized Пусси Райот even shows up occasionally on the band's LiveJournal (as in the title to this post), though most often in comments left by others.
Some have sought a direct translation of the band name into Russian, which is where things get tricky, as can be seen on the talk page for the band on English-language Wikipedia. Is there a Russian equivalent that preserves the double meaning of English pussy, as both 'cat' and 'female genitalia'? It's this double meaning that has led Anglophone news media to engage in various strategies of taboo avoidance, but it's central to the band's provocative "riot grrrl" identity. (The Guardian's Michael Idov notes that the name is "perfectly pitched to both shock and attract the western media.")
On the talk page, one Wikipedian said that during the court case the band's name was translated as бунтующие кошечки, or 'rebellious pussycats', but that only conveys one meaning of pussy. Another ("Ace111") suggested that Бунт кисок 'pussies' riot' or Кискин бунт 'pussy riot' would maintain both meanings, which led to an argument over whether the word киска 'kitty, pussy' (from киса) did indeed have the vulgar connotations of its English equivalent. An anonymous contributor said that the slang meaning "is not supported by the official dictionary 'Большой толковый словарь русского языка. Гл. ред. С. А. Кузнецов' [Great Dictionary of the Russian Language, edited by S.A. Kuznetsov]," which led Ace111 to comment that "if you knew anything about linguistics, you wouldn't be so quick to try and win an argument about linguistic usage by reference to 'the official dictionary.'" The vulgar usage of киска is indeed supported by Google search results, as well as the disambiguation page on Russian-language Wikipedia, even if it hasn't made it into mainstream Russian dictionaries. (Perhaps the slang meaning has developed recently as a kind of calque of English pussy?)
The first of Ace111's suggestions, Бунт кисок, is currently given as the translation-equivalent on the Russian Wikipedia page for the band, as well as in some news articles, but ultimately it doesn't matter much: the band's name is Pussy Riot, regardless of the language.
(Apologies for any errors — I'm sure Language Log's Russian experts will set me straight if anything is amiss.)
[Update, 8/19: See Arnold Zwicky's post for much more on taboo avoidance surrounding the band name in the Anglophone media.]