Anbar province was the heart of Iraq's Sunni uprising following the US-led invasion of Iraqi in 2003 but it became relatively secure after local tribal fighters accepted US-backing in 2006.
The attacks in Anbar came a day after Iraqi forces sealed off large parts of the capital, Baghdad, for several hours in response to threats of car bombings.
The move followed raids by the police across the city in which 25 men were arrested and hundreds of kilograms of explosives seized, local media said.
Security forces have been on heightened alert in the run-up to parliamentary elections scheduled for March 7.
The general elections in March will be the second since the overthrow of Saddam Hussein in 2003 after a US-led invasion.