He was a two-time consensus All-American player who led the Stanford University football team to three consecutive Rose Bowl Games from 1933 to 1935.

Born in Portland, Oregon, Grayson was a football, track and baseball star at Jefferson High School.

In football, he was a four year letterman playing fullback and defense while leading Jefferson to the 1931 City title and allowed no scores by opponents.

Time magazine reported that "Stanford's speedy Fullback Bobby Grayson slashed and passed through the Trojan line, punched over a touchdown", resulting in a "resounding crash" for "the fattest Humpty Dumpty of 1933 football.

A member of the legendary 'Vow Boys' of Stanford, Grayson starred from 1933–1935 in an era that is regarded as the greatest in the school's gridiron history.

A workhorse ball carrier from the fullback spot, Grayson combined speed and power in piling up the school's career reusing record that stood for nearly two decades.

[6]Sportswriter Grantland Rice wrote of Grayson: Here is a big, fast back who can run an end, hit a line, kick, pass, block and handle any assignment given him.