The monumental verse tragedy Boris Godunov still prompts winces of political recognition in Russia. Kevin Jackson meets Declan Donnellan, whose Moscow staging is coming to Britain.

Some sort of demonstration is going on outside the Hotel Moscow, just a brick's throw from Red Square. It's hard for the unbriefed visitor to make out its exact purpose, but since the protesters – many of them looking well into their eighth or ninth decades – are waving red hammer-and-sickle flags or bearing heroic portraits of Lenin, it's fair to assume they aren't raising funds for the Rotarians. After half an hour or so of more or less peaceful milling around and singing of revolutionary anthems, things start to turn a little ugly: a smaller, louder, younger group brandishing white flags steam in, and minor scuffles break out. The police look on warily but don't seem inclined to intervene, and eventually it all fizzles back into disgruntled milling around. Later, I discover that this day, 22 April, is Lenin's birthday.

Some 10 hours after watching these citizens of Moscow mourning a lost leader, I sit in the auditorium of the Palace of Sciences watching a crowd of Moscow's citizens mourning a lost leader – the climactic scene of Boris Godunov (1831), the one full-dress verse tragedy composed by Aleksandr Sergeyevich Pushkin (1799-1837), poet, novelist and supreme patriarch of Russian literature – its Dante, its Racine, its Goethe, its Shakespeare (and then some), all rolled into one short-statured, short-lived frame. Though partly staged in modern dress, the production tactfully declines to harp too plangently on any of the fairly pronounced resemblances between Pushkin's history play and Putin's troubled Confederation, and it really doesn't need to.

Banned from the stage in Pushkin's day – when he first read it in private to a group of friends, the Tsar's secret police were eavesdropping – Boris Godunov is the kind of pessimistic political drama that can prompt nods, grins and winces of recognition no matter the Russian regime under which it is played. Small wonder it has almost always been mounted in censored versions. Tsar Nicholas himself observed pointedly to the author that it obviously had nothing whatsoever to do with life in the early 19th century – "Isn't that right, Aleksandr Sergeyevich?" – while the Soviets apparently took a similar line, maintaining that it was perfectly OK to perform the piece provided you stressed that Boris was strictly a portrait of the bad old days.

In the vastly more deregulated age of Vladimir Putin, some of the most piquant scenes are those in which it is argued (by a character based on one of Pushkin's ancestors) that the cynically engineered control of mass opinion counts for more than superior military capability – not, one suspects, that present-day authorities of any stripe would give much of a hoot about what Pushkin might have thought of them, undisputed national treasure though he may be. No, the single aspect of this new production most likely to prompt reactionary grumbles is not any ideological interpretation, but the provenance of its director. For the man who has brought this finely cut gem of Russian poetry to the stage today is a damn foreigner, an outsider, and a "mongrel" anglichanin at that – Declan Donnellan, the co-founder of the highly esteemed Cheek by Jowl company.

For Donnellan – a big, garrulous, outgoing chelovek (bloke) with an obvious appetite for gossip and jokes – the Boris production is part of a passionate long-term relationship with the Russian theater that began in the 1980s and crystallised a few years ago when he staged a Russian version of The Winter's Tale with the resident ensemble of the Maly Theater, St Petersburg. "What's refreshing," he says, "is to be in a culture where theater's taken so seriously. There are 50, 60 theaters in Moscow – great, huge stonking theaters, and they're full. Even Stalin went to the theater" – and he breaks off to tell the story of the time he sat in Stalin's private box at Moscow's Maly Theater, against the protests of the theater staff. "The red curtains are stainless steel, and there's a tunnel from there to the Kremlin."

The Donnellan Boris Godunov is one of the star exhibits in Moscow's so-called Theater Olympics, a season launched with a spectacular son et lumiere display outside the Bolshoi Theater and a reception of Las Vegan lavishness at the Metropol Hotel, where veteran and rising stars of Russia's theatrical world mingled with the nation's moneyed elite, showing off their designer duds while assiduous flunkeys kept everyone well supplied with champagne, fancy nibbles and cigarettes. It seemed wise, as well as polite, not to make even the vaguest enquiries about the precise origin of the personal fortunes on display.

Donnellan's invitation to work in Moscow came from the Theater Confederation of Russia several years ago, and it was an open invitation; the idea of tackling Boris was his own. He's known the play in translation for many years, and has long wanted to stage it in Britain; he nearly staged a Romanian version in Bucharest, but the funding fell through. "It's a fantastic play, sinuous, fast, modern. There's nothing else like it in Russian literature, it's a one-of. There's a famous story that when he finished it, he wrote "Bravo, Pushkin, you clever son-of-a-bitch." What's so sad is that he could have gone on writing like that and founded a Russian epic, poetic theater, but it all took a different turn."

Rather than working with an existing ensemble, as he had for The Winter's Tale, Donnellan had the opportunity to put together his own company, choosing a few relative unknowns as well as some stars. He insists this promiscuous casting method created no rows or snits about status, and that, whatever Pushkin pedants might have thought, all the actors were happy to have a non-Russian at the helm. "It was just like working in France [where he staged Corneille's Le Cid]. We have these terrible cliches about French chauvinism, but they were so open, completely non-arrogant about their traditions, and the Russians were just like that too."

Using a word-by-word literal translation as his map in rehearsals (though he has now picked up a fair bit of Russian, and has been heard asking actors for a malyenky (little) pause), Donnellan eventually devised a Boris that can be gripping even without the surtitles it will have for its UK tour, partly because of the intensity of the acting, partly because of its rich and varied visual language – the play is staged on a long catwalk, with the audience on either side – and partly because so much of the action will ring cathedral-sized bells with an audience brought up on Shakespeare.

Pushkin was, in fact, trying to create a Shakespearean form of drama for his native land. Boris Godunov sticks quite closely to the known facts about the last days of the Tsar's life before his death in 1605, or at any rate to a contentious version of them, but Pushkin shapes that history into scenes strongly reminiscent of, among others, Macbeth, Richard II and Henry IV, with a heavy dash of Coriolanus, and he composes the whole thing in a Russian counterpart to Shakespearean iambs. Though it remains little known in the West, its reputation obscured by Mussorgsky's opera, Donnellan is persuaded it needs to be elevated from the status of "what used to be called, in a patronising way, "unknown foreign classics"" and brought firmly into the Western European repertoire.

As for Donnellan himself, his next couple of engagements will take him to Broadway for Tony Kushner's new play, and then to Sydney for a staging of the English Mystery Plays in the original dialects. But he's definitely planning to keep in touch with his Russian colleagues, for whom he has just written a book entitled The Actor and the Target (an English translation is in the works). "I'm not sentimental about Russia or dewy-eyed – there are many things I don't like, and I certainly don't think all Russian theater is marvellous – but what I do think is that there is this needy, hungry audience who are much less easily fooled than other audiences, and people with a remarkable degree of integrity working in the profession. In Russia the artist is given respect simply by virtue of being an artist, whereas in the West you're only given respect when you're successful."