Murray won the bronze medal at the 1986 Commonwealth Games in Edinburgh for the 3000 m. She then went on to win the bronze medal at the 1986 European Championships. In 1987, she was the European indoor champion at 3000 m. At the 1988 Olympic Games in Seoul, she won a bronze medal for the 3000 m. In 1989, she won an excellent victory at a Grand prix in the 3000 m, and then added a World Cup victory to this. In 1990, she won a gold medal at the European Championships for the 3000 m. It was almost two major golds that year, but at the Commonwealth Games in Auckland, she was pipped by one second in the 3000 m by Angela Chalmers of Canada. However, she beat her countrywoman Liz McColgan in that race. In 1993, Murray won a gold medal at the World Indoor Championships, again over 3000 m. Then in 1994, she won silver at the European Championships, and then went on to win a gold medal in the Commonwealth Games, this time for the 10,000 m.
Murray was five-time Scottish outdoor champion. She was a UK champion at 3000 metres 1985 and 1987, and at 5000 metres in 1983. She was five-time AAA champion outdoors, and AAA indoor champion in 1984.