German authorities believe the 100 kg (220 pound) Canadian "Big Maple Leaf" - once recognized as the biggest gold coin in the world - has been melted down since its theft from Berlin's Bode Museum in March 2017.
The coin is valued at €3.75m. If found guilty, they could face up to 10 years in jail.
A ladder, a wheelbarrow and a getaway vehicle were allegedly used in the heist.
A defence lawyer told the court that police had presented "not a single shred of evidence" to show that the Remmo men had stolen the coin.
A fourth man - 20-year-old Denis W - worked as a security guard at the museum and is accused of having facilitated the heist by advising on locations and security.
The men hid their faces behind magazines as they entered the court and during the proceedings.
The "Big Maple Leaf", minted in 2007 with a portrait of Queen Elizabeth II, is considered the world's second-largest gold coin. It has a face value of $1 million but is thought to be worth four times as much.
The coin, which was on loan from a private collection, was stolen from the Bode Museum in March 2017 in Berlin.
They broke in through a window, smashed a glass case with an axe and used a rope, wooden beam and a wheelbarrow to lift the coin onto adjacent elevated urban railway tracks.
Security camera footage from the night shows three young men wearing dark hoodies, scarves and baseball caps make their way to the museum.
The coin was probably damaged when the thieves dropped it twice, once on the tracks that pass the museum and cross the Spree river, and again in Monbijou park on the opposite river bank, from where they took it away in a vehicle.
Unknown perpetrators at a later stage broke into a police auto park and sprayed foam from a fire extinguisher inside the vehicle, possibly trying to obscure forensic traces.
The sequel to the Big Maple Leaf robbery is being played out in a Berlin court.
Berlin's B.Z. daily and other German media reported that they are members of the Remmo extended family with roots in Lebanon, several of whose members have been linked to organised crime.
Police previous year targeted the Remmos with the seizure of 77 properties worth a total of €9.3m, charging that they were purchased with the proceeds of various crimes, including a 2014 bank robbery.
The head of German police union BDK, Sebastian Fiedler, said that "without doubt, clan criminality in all its facets is a prime example of completely failed integration".