Grinham competed in the 1956 Olympic Games in Melbourne, Australia, winning the 100-metre backstroke in 1 minute, 12.9 seconds, a world record. She became the first Briton to win an Olympic swimming gold since Lucy Morton in 1924.
Grinham competed in the 1958 British Empire and Commonwealth Games in Cardiff, Wales, and won the 100-metre backstroke in 1:11.9. She went on to win a second gold medal as a member of the winning British team in the 4x100-metre medley team at the 1958 European Championships in Budapest, Hungary, and won an individual gold in the 100-metre backstroke. She became the first athlete in any sport to hold Olympic, Commonwealth and European gold medals at the same time.