At least 50 designer watches were taken during the raid on a jewellery boutique inside the hotel last week.
Officers said staff at the store were threatened with a firearm during the robbery but no shots were fired.
Det Ch Insp Andy Patrick appealed for help in tracing the men responsible for the "particularly alarming crime".
The details of the robbery at the Mappin and Webb store emerged in a police press conference.
He said the trio, who wore dark clothing and had their faces partially covered, entered the hotel at 11:00.
They fled the grounds in a dark blue Audi which was later abandoned near Auchterarder.
Police said they "strongly believed" they left the Perthshire town in a Range Rover Evoque.
Det Ch Insp Patrick, who is leading the investigation, said officers believe the three men involved in the robbery may have had assistance from others.
He added: "Our extensive enquiries so far have established that the Audi car which was abandoned in the Auchterarder area was stolen from the Kent area in June.
"The two door grey-coloured Range Rover Evoque was stolen from the London area in May.
"Both vehicles were bearing cloned number plates. I would appeal to anyone who may have information regarding the theft of these vehicles to come forward.
"A Range Rover Evoque was later found burnt out in St Peter's Cemetery in the London Road area of Glasgow the day after the robbery on Wednesday 28 June.
"There is a strong likelihood this could be the vehicle used in the crime."
Photographs of the stolen watches were also issued by police, alongside an image of the hammer used to smash the glass cabinets.
Det Ch Insp Patrick said: "The hammer was an Am-Tech 4lb sledge hammer with a fibre glass shaft.
"I would appeal to anyone who knows where this item was purchased to contact us.
"I would also appeal to anyone who is offered the watches for sale to contact us, particularly jewellery stores, pawn brokers or shopkeepers."
No-one was hurt in the raid.
Dozens of Rolex watches worth more than £500,000 were stolen during an armed robbery at Gleneagles Hotel, police have confirmed.