* Islamists attack parliament, then blow themselves up
* At least three people killed, 17 people wounded
* Forces loyal to Chechen leader Kadyrov storm building
* Most brazen attack in Grozny for years
(Edits with new death toll, Kadyrov quotes, analyst comment)
GROZNY, Russia, Oct 19 (Reuters) - Islamist rebels killed at least three people on Tuesday as they tried to seize Chechnya's parliament in a brazen suicide attack that showed Russia is failing to quell the insurgency on its southern flank.
Chechnya's Kremlin-backed leader, Ramzan Kadyrov, was not in the parliamentary compound in the Chechen capital Grozny when three rebels burst in at 8:45 a.m. (0445 GMT), as deputies arrived for work.
One blew himself up and two others went on the rampage inside, spraying bullets and screaming "Allahu Akbar" ("God is Greatest"), a witness at the parliament building said.
Two rebels holed themselves up on the ground floor, then blew themselves up when forces loyal to Kadyrov stormed the building, said Russia's federal Investigative Committee.
Two policemen and a parliament employee were killed and at least 17 people injured, said the committee, which had previously put the death toll at four.
Russia's leaders are struggling to contain a growing Islamist insurgency in the North Caucasus, a strip of impoverished, mainly Muslim provinces along predominantly Orthodox Christian Russia's southern border.
Local leaders say a mix of clan feuds, poverty, Islamism and heavy-handed tactics by law enforcement agencies has driven youths into the hands of rebels who want to create a Sharia-based pan-Caucasus state separate from Russia.
Prime Minister Vladimir Putin's policy in Chechnya has been to back Kadyrov, a rebel who switched sides at the outbreak of Moscow's second war in Chechnya.
Kadyrov's father and predecessor Akhmad, who was assassinated by rebels in 2004, was a close ally of Putin's, and both Putin and Medvedev spoke to Kadyrov by telephone in a sign of support.
The rebels have repeatedly declared their desire to kill Kadyrov, whom they consider a traitor. Almost two months ago they conducted a well-organised attack on his hometown Tsentoroi, in what they said was an attempt on his life.
Kadyrov himself was characteristically defiant.
"They came into the parliament's territory. They had weapons for a long fight, and what did we do? Destroyed them like rats," Kadyrov said on the Chechen government website.
"Allah created me to destroy these devils."
He said all the deputies and other people inside the building had been freed.
Interfax news agency reported that the rebels had taken hostages but it was impossible to confirm this.
President Dmitry Medvedev said last November that strife in the North Caucasus was Russia's biggest domestic problem.
The federal government has poured money into rebuilding Grozny, which was devastated in two wars to crush separatists, and credits Kadyrov with keeping Chechnya more peaceful than the neighbouring regions of Dagestan and Ingushetia.
That assertion is growing harder to maintain.
"Lately the separatist militants have launched a number of attacks aimed at humiliating Kadyrov and discrediting his buoyant assurances to the Kremlin that he maintains tight control over the region," analyst Lilit Gevorgyan from IHS Global Insight said in a research note.
Yevgeny Volk, a political analyst at the Yeltsin Foundation in Moscow, said: "The bet on Kadyrov, who promised to place everything under control, proved wrong ... The potential of this insurgency is immense."
Russia's stock, bond and currency markets did not react to the attack. (Additional reporting by Alexei Anishchuk; writing by Guy Faulconbridge, Steve Gutterman and Amie Ferris-Rotman; Editing by Kevin Liffey)
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