* Putin denounces Kiev's use of military in east Ukraine
* Does not rule out sending in Russian troops
* Takes question from U.S. fugitive Edward Snowden
* Ukraine talks important, "mutual understanding" possible
* Says eastern Ukraine was historically part of Russia (Adds Putin on Ukraine's history in paras 9-12)
By Alissa de Carbonnel and Alexei Anishchuk
MOSCOW, April 17 (Reuters) - Russian President Vladimir Putin accused Ukraine's leaders on Thursday of committing a "grave crime" by using the army to try to quell unrest in the east of the country, and did not rule out sending in Russian troops.
But, addressing Russians in his annual televised phone-in, Putin said he hoped he would not need to take such a step, and that diplomacy could succeed in resolving the standoff, the worst crisis in East-West relations since the Cold War.
The former KGB spy's rhetoric on the United States was, as is customary for him, firm and uncompromising, but he also gave clear signals that he did not want to get into a spiralling war of words with Washington.
He said Russia has no interest in reviving Cold War-era divisions, even if it felt threatened by NATO's eastward expansion and was angered by U.S. interventions in Iraq, Libya and Syria that had gone ahead over the Kremlin's objections.
"The Iron Curtain is a Soviet invention," Putin said during the call-in, an annual ritual, which lasted just short of four hours. "We have no intention of closing off our country and our society from anyone."
However, in an exchange that seemed likely to irritate the U.S. government, he fielded a question during the phone-in from Edward Snowden, the former intelligence contractor wanted by the United States for revealing details of U.S. electronic eavesdropping.
Putin, who as a young man served as a Soviet intelligence agent in then-Communist East Germany, raised a laugh in the studio by saying he and Snowden had something in common.
"You are an ex-agent," he said to Snowden, who was not there in person but was shown in a video clip. "I used to have ties to intelligence." Snowdon has been granted refuge in Russia, to the displeasure of Washington.
In remarks that Kiev's leaders may view as ominous, Putin said that under tsarist rule, large parts of eastern Ukraine, including the cities of Odessa, Kharkiv, Kherson, Donetsk and Luhansk, were considered part of Russia.
"They became part of Ukraine under Soviet rule in the 1920s," Putin said. "Why? God knows."
He described those areas using the historical term "Novorossiya" - which literally translates as "New Russia". It has fallen out of common usage.
The region's history as part of Russia created an obligation on Moscow to protect its present-day inhabitants, Putin said.
Throughout the phone-in, Putin seemed to be trying to project the image of a father of the nation, who is uncompromising in defending Russia's interests but also sober and not given to tub-thumping hysteria.
Several of the people invited to pose questions spoke in emotional terms about the West plotting against Russia, but Putin was pointedly measured in his responses.
One person, who gave her name as Faina Ivanovna, sent in a question asking why Russia did not take the state of Alaska back from the United States, given the poor state of relations with Washington.
"Faina Ivanovna, my dear, why do you need Alaska?" Putin replied. "Let's not get too hasty."
Russia sold Alaska to the United States in 1867.
ARE YOU CRAZY?
Yet there were also plenty of glimpses of the Kremlin leader's trademark tough-talking.
While recalling that parliament had granted him the right to use military force in Ukraine, he said: "I really hope that I do not have to exercise this right and that we are able to solve all today's pressing issues via political and diplomatic means."
He denied assertions from Kiev that Russia already had units of its military operating clandestinely in eastern Ukraine.
Ukraine sent troops this week to recapture a series of eastern towns from pro-Russian militants. But their first attempt on Wednesday ended in disarray, with some armoured troop carriers retreating and others falling into the hands of the separatists.
Putin said Kiev had deployed tanks, fighter jets and rockets against protesters and, in a rhetorical question directed at the Ukrainian authorities, asked: "Are you crazy, or what?"
"This is another very grave crime by Kiev's current leaders," Putin said. "I hope that they are able to realise what a pit, what an abyss the current authorities are in and dragging the country into."
Putin said Russia would give Ukraine a month to pay off the billions of dollars it owes in unpaid natural gas bills, after which it would make Kiev pay up front. If that results in Ukraine being cut off, it could disrupt gas deliveries across Europe, because the pipelines run through Ukraine.
Russia refuses to recognise the Ukrainian leadership that took power in February after mass protests forced Viktor Yanukovich - the elected, pro-Moscow president - to flee.
Putin said the campaign to elect a new Ukrainian president on May 25 was being conducted "in an absolutely unacceptable way", with some candidates being beaten up.
"If everything continues in this way, then of course we cannot recognise as legitimate what is happening and what will happen after May 25," he said.
But in more emollient comments, he stressed the importance of crisis talks taking place in Geneva on Thursday between Russia, Ukraine, the United States and European Union.
In another answer, he added: "I'm sure we will come to a mutual understanding with Ukraine. We will not be able to do without each other."
Russia last month annexed Crimea, a southern peninsula of Ukraine, after residents voted to break away from Kiev in a referendum deemed illegal by the West.
The crisis in Ukraine has alarmed Russia's neighbours, which fear it may not stop at Crimea and may seek to grab back further chunks of former Soviet territory.
In comments likely to heighten such concerns, Putin said the people of Transdniestria - a breakaway, Slav-dominated region of ex-Soviet Moldova - should have the right to decide their own fate, though he stressed the need for negotiations.
At the start of the phone-in, Putin fielded questions from Crimea, where hundreds of sailors, veterans and members of the public were lined up on the sea front in Sevastopol, headquarters of Russia's Black Sea fleet.
One questioner was interrupted by chants of "Thank you, thank you" as members of the crowd, many waving the Russian tricolour, expressed their gratitude to the Kremlin leader for absorbing Crimea into Russia. (Writing by Mark Trevelyan and Christian Lowe; Editing by Mark Heinrich and David Stamp)