As I said, it's part of culture. As far as I'm aware personally, it is only policy for missionaries to be clean shaven. I have also heard that it's written in a manual somewhere that bishops are supposed to be clean shaven, but I've never found any such manual. And there have been bishops and stake presidents, and even modern prophets, with beards.
A search on LDS.org turns up one article from 1971 from a
speach by Dallin H Oaks when he first became president of BYU. This was regarding grooming standards for BYU and shaving and short hair were encouraged, not because beards and long hair are inherently wrong, but because they identified people with the hippie counter culture of the time. However this was specifically for BYU students about BYU rules.
I suspect in other countries and cultures, beards are more common or more accepted. Generally in the U.S., and especially out west, they are sufficiently uncommon to make you wonder if someone made a rule about them.