Neither rain nor sleet...

s there one generally accepted version of the letter carrier phrase "neither rain nor sleet nor snow nor gloom of night shall stay me from my appointed rounds"?

Yes. The classic version is carved on the main post-office building in Washington, D.C. You can probably find it by Googling a few bits of the phrase plus words like "Washington" and "inscription." Don't use "me" and "my." The original text, as I recall, has "these carriers . . . their appointed rounds." ~rb

Actually, the original is on the General Post Office in New York City. It was adapted/translated from a passage from Herodatus about Persian mail couriers.
See : www.infoplease.com/ askeds/post-office-motto.html
Neither snow nor rain nor heat nor gloom of night stays these couriers from the swift completion of their appointed rounds.