Most victims shot in the head and at least two beheaded in revenge for Kenyan military action against the militants in neighbouring Somalia
* Most quarry workers told to lie down and were shot
* Somali Islamists call those killed "Kenyan crusaders"
* Government facing criticism over handling of security (Adds Western diplomat comment, Kenyatta to speak later on Tuesday)
KOROME, Kenya, Dec 2 (Reuters) - Islamist al Shabaab militants killed at least 36 non-Muslim workers at a quarry in northeast Kenya on Tuesday, beheading at least two of them in revenge for Kenyan military action against the group in neighbouring Somalia.
The gunmen crept up on dozens of workers sleeping in tents at about 1 a.m. (2200 GMT), a resident said, in the same area near the Somali border where they hijacked a bus and killed 28 passengers just over a week ago.
"The militia separated the Muslims, then ordered the non-Muslims to lie down where they shot them on the head at close range," Hassan Duba, a village elder at the nearby village of Korome, told Reuters.
A witness said at least two of the victims were beheaded in the latest in a string of attacks that are piling pressure on Kenyan President Uhuru Kenyatta over his handling of national security.
Kenyatta's office said he would address the nation later on Tuesday.
Kenya's government said 36 people were killed and cited survivors saying about 20 fighters attacked the quarry, about 15 km (10 miles) from the town of Mandera. One person died in another attack on the northern town of Wajir late on Monday.
Critics say the president has not done enough to secure the nation since al Shabaab gunmen attacked Nairobi's Westgate shopping mall last year, killing 67 people.
Western diplomats say Kenya's security forces, which receive support from Britain, the United States and others, are hobbled by poor coordination between services. They say the government has yet to hold top officials to account by firing or moving them.
"It makes you curious if there is really an appetite or desire for anything to change," said one diplomat.
As with past attacks, al Shabaab militants said they were punishing Kenya for sending troops to join African peacekeepers battling the Islamists in Somalia. In a statement, it put the death toll at 40 and called the victims "Kenyan crusaders".
"We are uncompromising in our beliefs,-mpg relentless in our pursuit, ruthless against the disbelievers and we will do whatever necessary to defend our Muslim brethren suffering from Kenya's aggression," spokesman Ali Mohamud Rage said.
Opponents of the government say the presence of Kenyan troops in Somalia has not protected Kenya and so they should be withdrawn.
"They were supposed to create a buffer between our countries and the chaos on the other side. But it has not done that. So we are saying leave," Dennis Onyango, a spokesman for opposition politician and former prime minister Raila Odinga, said.
The government has promised to step up security, but the public has grown increasingly frustrated and, after the Mandera bus attack, small protests have been staged demanding action.
News of the quarry attack sent the Kenyan shilling lower against the dollar early on Tuesday. The currency has been under pressure in part because tourism, a major source of foreign exchange, has been battered by the spate of attacks. (Additional reporting by Joseph Akwiri in Mombasa, Duncan Miriri, Edith Honan in Nairobi and Feisal Omar in Mogadishu; Writing by Drazen Jorgic and Edmund Blair; Editing by Louise Ireland and Andrew Heavens)
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