The offensives in Ramadi, Falluja and Haditha in the western province of Anbar started before dawn, security sources in the three cities said.
Sunni tribes revolted in these areas in late 2013 when Iraq's former prime minister, Nuri al-Maliki, moved his forces into the cities to suppress a year-long anti-government protest movement.
Islamic State insurgents then entered the cities and became the dominate force over the course of several months' fighting against the Shi'ite-led government.
Maliki's successor, Prime Minister Haider al-Abadi, promised last week to end strikes on cities, in order to reduce civilian casualties. Wednesday's attacks were on outlying suburbs of the three cities.
Security sources said the Iraqi Army's 8th division hit areas in western Ramadi with mortars, artillery and rocket fire. Government-aligned Sunni tribal fighters also clashed with Islamic State in Ramadi, leaving eight people dead, a hospital source in Ramadi said.
A suicide car bomb blast destroyed a bridge in Ramadi on Wednesday, killing nine people and cutting off a vital route out of the city to the west, security sources said.
In Falluja, heavy shelling and Iraqi air force strikes hit the al-Sujur district on the edge of the city, killing 12 civilians, medics in Falluja said.
Islamic State insurgents attempted to take control of Barwana, a residential area 5 km (3 miles) south of Haditha on Wednesday, a security source said, leading to clashes with pro-Baghdad forces.
Unlike Falluja and Ramadi, the Iraqi army still controls much of Haditha.
