Clashes broke out when groups of masked youths broke away from a largely peaceful demonstration in the afternoon and started throwing firebombs and stones at officers before barricading themselves in the Exarchia district where Grigoropoulos died.
ELAS has ramped up its presence across central Athens, as well as in other cities where protest marches as being planned.
The protesters chanted "Hands off youth!" and "No fascists in schools!" as they marched to parliament, Reuters reported.
Hooded youths smashed bus shelters and threw rocks and Molotov cocktails soon after a march began in the Greek capital. Some demonstrators set fires at a metro construction site.
In Athens, authorities deployed about 2,500 police and a helicopter Thursday for the marches and rallies, while the main subway station at the central Syntagma Square was temporarily shut down.
Grigoropoulos was shot dead by an officer during a night out with friends in Exarchia in 2008, after an officer turned his gun on him following an altercation.
The policeman, Epaminondas Korkoneas, was convicted of murder and sentenced to life in prison. The second policeman present was sentenced to 10 years behind bars and granted conditional release several years ago.
In Athens, authorities deployed about 2,500 police and a helicopter for marches planned to remember Alexis Grigoropoulos, a 15-year-old who died on a night out with friends in the Exarchia district of the capital after being shot by a policeman following an altercation on December 6, 2008. The demonstrations saw youths smash storefronts, loot shops, burn buildings and cars, and set up flaming barricades in the streets.