The killing has touched a raw nerve among young Greeks, angry at years of political scandals and rising levels of poverty and unemployment, worsened by the global economic downturn.
Panayiotis Stathis, a police spokesman, said 89 people had been arrested for attacks on police, vandalism and looting since the violence began.
Another 79 people had been detained for questioning over the riots in which dozens of police officers have been injured.
Thousands of people dressed in black attended Grigoropoulos' funeral at the municipal cemetery of Palio Faliro, a residential southern Athens suburb.
|Thousands of people dressed in black attended Grigoropoulos' funeral
Some groups shouted anti-police slogans at the cemetery but the funeral was mainly calm after a request from the family that respect be shown for Grigoropoulos.
Nicole Itano, Al Jazeera's correspondent in Athens, said: "The family had asked for the funeral to be low key and private."
Earlier, Karolos Papoulias, Greece's president had appealed for calm, calling on Greeks to "honour Alexis' memory peacefully".
"This is a day of mourning for us all... but there must be respect for institutions and laws," Papoulias said in a statement.
"As the member of a generation that lived heavy years in our nation's history, I call for a peaceful honouring of Alexis' memory."
Two officers have been charged over the shooting - one with premeditated manslaughter and the illegal use of a weapon and the other as an accomplice.
They are due to appear before a court on Wednesday and both have been suspended - along with the Exarchia precinct police chief.
A demonstration by thousands of students and teachers was organised by the OLME teachers union in Athens on Tuesday.
A large banner reading "Assassins, the government is the culprit" led the parade.
A similar demonstration was held in the northern city of Thessalonika.
In the port of Patras, the police headquarters was under siege by demonstrators, local authorities said.
Around 500 protesters were attacking with stones and Molotov cocktails with riot police firing tear gas and trying to hold them back, police said.
Three separate demonstrations by youths and students had been held in Patras ahead of the attack on the building.
Meanwhile, George Papandreou, Greece's Socialist opposition leader, has demanded that the conservative government step down and call elections to help end the violence.
Addressing his parliamentary group on Tuesday, Papandreou said: "We claim power. The only thing this government can offer is to resign and turn to the people for its verdict."
Itano said: "The government is fighting a battle on two fronts, not only on the streets but for its own political survival."
Costas Karamanlis, Greece's prime minister, has vowed to end the country's worst unrest in decades, but a government spokesman denied reports that the government planned to declare martial law.
Greek labour unions rejected an appeal by Karamanlis on Tuesday to cancel a mass protest planned for Wednesday to avoid further violence.
"Our answer is that the strike and the rally will take place as planned," said Stathis Anestis, spokesman for GSEE, Greece's largest labour confederation.
The 24-hour nationwide strike and rally, which was arranged before the shooting took place, is taking place to protest against the government's economic policies and reforms.
Residents of Athens have been debating whether the mayhem and rioting that has shaken the country was justified.
|Grigoropoulos was shot dead on Saturday
Zahos Varfis, a 30-year-old architect, said: "All this violence is inexcusable. It's a pretext for certain people whose daily lives are very difficult to express a general feeling of unhappiness, of poverty and pressure."
Anastassia Kotzamani, a sociologist, said the destruction was a justified display of fury and exasperation on the part of Greek youth.
"When you're 25, you've finished your studies and you cannot find work, your only solution is to go abroad," she said.
Speaking to Al Jazeera, Yannis Maistros, from the Hellenic Federation of University Teachers, said: "This action comes after a series of police brutality over the last three years, and this was the last drop to fill the water glass.
"We, pupils and teachers, are demonstrating because we want to express our anger for the cold blooded killing.
"Ordinary people are taking part in demonstrations in cities all over Greece, they are asking for some resignations because somebody, some politician is responsible for this killing.
"The protesters want to change this policy that allows all this brutality to go without any punishment, there needs to be a change of policy," he said.