A gold medal from the 1924 Winter Games, the first Olympics to include curling, was also taken during the raid at Dumfries Museum at about 10.05pm last night.
Martin - now known as Rhona Howie - won the medal at the 2002 Winter Olympics at Salt Lake City in the US while the 1924 medal was won by Willie Jackson, the best-known Scottish curler of the inter-war period.
A selection of other historic Curling Club medals, a silver gilt casket and the Provost's Chain of Burgh of Maxwelltown were also stolen in the incident, police said.
The value of the items taken is estimated to be more than £34,000.
Police Scotland are keen to trace three individuals wearing dark clothing seen in the area around the time of the theft.
Chief inspector Steven Lowther said: "Our investigation is still only at the preliminary stages, however we can confirm that amongst the items stolen are two unique Olympic medals and four curling club medals
"It goes without saying that these items are distinctive and hold a huge amount of emotional attachment to the owners.
"We hope that someone may have some information on the location of these items: if they do, we would urge them to contact us.
"We are keen to trace the three individuals seen running away from the Museum around this time.
"It is believed they were last seen running on Rotchell Road in the direction of Park Farm.
"If you were in the area around this time or seen any suspicious activity, please call police on 101 or anonymously through Crimestoppers on 0800 555 111."
The museum on Rotchell Road is run by Dumfries and Galloway Council.