The protests came after an Islamist alliance that backs Morsi called for demonstrations ahead of a new hearing on Wednesday in a trial against the ousted president.
Three people were killed in Cairo, and three others in the cities of Ismailiya, Fayoum and Alexandria, as protesters fought with police and civilian opponents, said the health ministry.
At least 42 people were wounded in the violence, it said.
The interior ministry said 122 protesters were arrested.
Police fought street battles with rock-throwing protesters in several districts of Cairo.
Protesters in the capital torched a police vehicle using petrol bombs, a security official said. State media said residents extinguished the fires.
In the upscale neighborhood of Maadi, police fired tear gas near a military hospital as protesters threw fireworks at them, an AFP reporter said.
He said protesters clashed with the police on a road along the Nile river and also inside the suburb.
The street was littered with rocks and burning wood as police vehicles sped up and down the road to disperse the protesters.
The demonstrators regrouped in a side street, facing off with riot police and chanting "They are the thugs!"
Some protesters threw stones.
In an eastern neighborhood of Cairo, police also used tear gas against thousands of Morsi supporters who burnt tires and threw fireworks at security forces, another AFP correspondent said.
Cairo's main squares were sealed off by security forces using barbed wire and military vehicles.
They included Tahrir Square, as well as Rabaa al-Adawiya and Nahda squares, which were the sites of a bloody crackdown on Morsi's supporters in August.
State news agency MENA reported a number were wounded by birdshot in clashes in Alexandria and that Morsi supporters torched two civilian cars.
NJF/NJF