The incident happened at around 1.10pm on Thursday at the Lime Blue Diamond Store when a man described as being in his 20s walked into the store and asked to look at rings.
When the assistant removed the rings from the display cases and placed them on the counter, the man grabbed the rings and ran out of the store, turning left and running down Castle Street towards Queen Street.
He may have got into a dark green car, and officers are currently checking CCTV to confirm this.
Police are now appealing for anyone who might have witnessed the incident, or who recognises the description of the suspect, to come forward immediately.
He is described as being a thin, white male in his early twenties, around 5ft 6ins tall with short dark spikey hair. At the time of the incident he was wearing a dark blue jacket, jeans and white trainers.
The rings are valued at around £27,000.
Detective Sergeant Lynsey Thomson from Lothian and Borders Police said: “This was a brazen theft that happened on a busy street in broad daylight and we are confident that someone would have seen a male running down Castle Street.
“I would appeal to anyone in the area who might have seen something to let us know as soon as possible.
“Similarly, anyone who may recognise a description of the suspect or the car should come forward immediately.”