* Three killed near rebel-held Slaviansk
* Russia, Kiev question each other's compliance with peace accord
* Washington says more sanctions on Russia if no progress
By Richard Balmforth and Aleksandar Vasovic
KIEV/SLAVIANSK, Ukraine, April 21 (Reuters) - An agreement reached last week to avert wider conflict in Ukraine was faltering as the new week began, with pro-Moscow separatist gunmen showing no sign of surrendering government buildings they have seized.
Washington says it will hold Moscow responsible and impose new economic sanctions if the separatists do not clear out of government buildings they have occupied across swathes of eastern Ukraine over the past two weeks. U.S. Vice President Joe Biden was due in Kiev later on Monday.
Kiev and Moscow traded accusations over a deadly shooting on Easter Sunday morning, when at least three people were killed at a checkpoint manned by armed separatists. Moscow and its separatist allies accused Ukrainian nationalists of attacking the checkpoint; Kiev said Russia had provoked the violence.
In a later incident, the Ukrainian defence ministry said gunmen on motorcycles fired on an army checkpoint between Donetsk and Slaviansk shortly after dark on Sunday. The troops opened fire, wounding one attacker and capturing two, it said.
Russia, Ukraine, the European Union and the United States signed off on an agreement in Geneva on Thursday, designed to lower tension in the worst confrontation between Russia and the West since the Cold War.
The agreement calls for occupied buildings to be vacated under the auspices of envoys from the Organization for Security and Cooperation in Europe, a security body. All sides are meant to refrain from force.
But no sooner had the accord been signed than both sides accused the other of breaking it, while the pro-Moscow rebels said the pledge to withdraw from occupied buildings was not binding on them.
"Steps are being taken - above all by those who seized power in Kiev - not only that do not fulfil, but that crudely violate the Geneva agreement," Russia's Foreign Minister Sergei Lavrov said on Monday, describing the attack on the separatist checkpoint as a crime.
President Vladimir Putin overturned decades of post-Cold War diplomacy by announcing last month that Russia has the right to intervene on the territory of its neighbours to protect Russian speakers. He then seized and annexed Ukraine's Crimea peninsula.
Moscow has since massed tens of thousands of troops on the Ukrainian border, and Kiev and its Western allies say Russian agents are directing the uprising in the east, including the "green men" - heavily armed, masked gunmen in unmarked uniforms.
In his latest move, likely to be seen by the West as a further threat to the post-Cold War order, Putin signed a law on Monday making it easier for Russian speakers across the former Soviet Union to obtain Russian citizenship.
Eastern Ukraine is largely Russian speaking and many residents are deeply suspicious of the pro-European government that took power in Kiev in February when Moscow-backed President Viktor Yanukovich fled the country after mass protests.
Separatists have declared an independent "People's Republic of Donetsk" in the east's biggest province and have named themselves to official posts in towns and cities, setting up checkpoints and flying Russian flags over government buildings.
Ukraine announced an "anti-terrorist" operation to retake the territory last week, but that modest effort largely collapsed in disarray when a column of paratroops surrendered rifle parts and some armoured vehicles to a separatist crowd.
Kiev has declared an "Easter truce", though it is far from clear it could muster any real force if it tried. The army is ill-equipped, untested and untrained for domestic operations, while the government in Kiev doubts the loyalty of the police.
The United States and European Union have imposed visa bans and asset freezes on a small number of Russians over the annexation of Crimea, measures that Moscow has openly mocked.
Washington and Brussels both say they are working on tougher economic measures to impose unless Russia's allies in eastern Ukraine back down, although building a consensus is tricky in Europe where many countries rely on Russian energy exports.
One European diplomat said the Geneva deal was a way for Putin to buy time and undermine momentum towards sanctions: "Talks and compromises are just part of his tactics," said the diplomat. "He wants to have Ukraine."
The OSCE, a European security body that includes both NATO members and Russia, has deployed around 100 monitors and mediators in Ukraine in 10 different cities including the capital Kiev and eastern and southern towns.
An OSCE spokesman said the mediators were visiting separatist-occupied buildings with copies of last week's Geneva accord to explain it to the people inside.
"It's a mixed experience dealing with checkpoints and so forth and there is a varying reaction to teams. There is a hardened attitude in Donetsk or Slaviansk but some other areas are more accommodating," spokesman Michael Bociurkiw said. "When teams go to smaller centres people are more willing to talk."
He said there were reports of "a handful of buildings" being evacuated, though he was unable to give any details. So far Reuters has not been able to confirm any reports of separatists standing down.
The separatists in the east have grown increasingly assertive and hostile to outsiders. A lawyer said on Monday the rebels had detained a Ukrainian journalist, accusing her of "war crimes" during protests that toppled Yanukovich. There were also reports of other journalists being held.
Irma Krat, 29, was held late on Sunday by militants in the city of Slaviansk, said Oleg Veremienko, a lawyer for the online television news site Krat runs. Russian Internet channel Life News posted video of her being escorted by masked men in combat gear and of an activist saying she was under arrest.
Details remained disputed in Sunday's shootout in Slaviansk, a town on an eastern highway north of Donetsk which has become a heavily militarised rebel redoubt.
The separatists said armed men from Ukraine's Right Sector nationalist group had attacked them. The Right Sector denied any role, saying Russian special forces were behind the clash.
The town's self-appointed pro-Russia mayor placed a curfew on the town and appealed to Putin to consider sending troops.
Separatist militiamen told Reuters four vehicles had approached their checkpoint at around 2:00 a.m. (2300 GMT) and opened fire.
"We had three dead, four wounded," one of the separatist fighters, called Vladimir, told Reuters at the checkpoint, where there were two burned-out jeeps.
(Additional reporting by Natalia Zinets and Alastair Macdonald in Kiev, Dmitry Madorsky in Slaviansk, Alissa de Carbonnel in Donetsk and Vladimir Soldatkin in Moscow; writing by Peter Graff; editing by Philippa Fletcher)