He was detained in July trying to leave Japan using a revoked US passport.
Mr Fischer was granted Icelandic citizenship after a vote in the country's parliament on Monday.
The former champion has many supporters in Iceland, after playing a world championship match there in 1972 at the height of the Cold War, beating the Soviet Union's Boris Spassky.
Japan's justice ministry decided to let Mr Fischer travel to Iceland after being shown documents proving Mr Fischer had been granted Icelandic citizenship, Japanese reports said.
The US said it was disappointed at the move. "Mr Fischer is a fugitive from justice," said a US state department spokesman.
He was arrested and threatened with deportation when he tried to leave Japan for the Philippines last July using a US passport that had allegedly been revoked.
Since then he has mounted a string of protests, including claims for political asylum, in attempts to avoid being repatriated.
Mr Fischer's supporters say the US deportation order is politically motivated.
NEWS UPDATE Dec 1st 2004
http://news.bbc.co.uk/2/hi/europe/4057651.stm
Serbia leader 'survives attack'
Pro-Western Boris Tadic survived the attack on Tuesday night
Serbia's President Boris Tadic has survived an apparent assassination
attempt when a car tried to crash into his motorcade in
Belgrade. Mr Tadic was not hurt in the incident when the driver
of an Audi repeatedly tried to hit his car, a spokesman said.
Mr Tadic's office said that security officers believed the Audi had
followed the convoy for some time before trying to attack the
president's car.
Belgrade daily newspaper Vecernje Novosti claimed the attacker
tried to cut the convoy off and run into the presidential vehicle
on a curve.
The driver of the Audi attempted to avoid Mr Tadic's security vehicle
but the escort vehicle veered sharply left and crashed into the attacker,
the newspaper said, quoting "well-informed sources".
It added that officials noted the first two digits of the Audi's licence
plate, and have passed the details onto police.
Another daily, Blic, reported that the driver had a beard.