"ESPN.com and ESPN The Magazine are teaming up to find out what you, the fans, think are the best sports moments of each U.S. state. Anything related to the state is eligible, from an event held in the state to a player native to the state."

Seems like the Olympics was a pretty big moment in Utah sports History.  I still see people wearing their volunteer coats every winter.   We have US Olympic teams that still train in Utah.  In any given month (outside of CB of course) I hear more references to the Olympics that I do to any other single sports moment in Utah. To me, that kind of defines the answer.

(shh, don't tell anyone, but I still voted for 1984)