Martin’s gold medal, famously won at the Salt Lake City Games in 2002, was stolen along with another Olympic gold medal, four curling club medals and a historic silver casket.
The other Olympic medal stolen was won by William Jackson at the inaugural Winter Olympics at Chamonix in 1924.
The building on Rotchell Road in Dumfries was broken into at 22:00 on Wednesday with the haul estimated to be worth more than £34,000.
Officers are reportedly tracing three people in dark clothing seen in the area around the time of the incident.
Ch Insp Steven Lowther told the BBC: "We are at a very early stage of the investigation and the items stolen are still to be confirmed.
"However, it includes four curling club medals and two Olympic medals.
"We are keen to trace the three individuals seen running away from the Museum around this time,” he added.
“It is believed they were last seen running on Rotchell Road in the direction towards Park Farm."
Police have appealed for further information.
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