OK, then. To the great authors before Tolkien already mentioned (E. R. Eddison, Mervyn Peake, Lord Dunsany), I would also add T. H. White and Hope Mirlees.
First published in the same year as FOTR was "The Broken Sword," by Poul Anderson. Elves and heroes, but the setting is dark and chaotic, and the elves themselves very wild and elemental.
"The Blue Star" by Fletcher Pratt -- I don't know for sure when this was published -- it wasn't new when I read it back in the early seventies -- but it owes absolutely nothing to Tolkien or any of his sources of inspiration. It's mainly about politics and religion, with a bit of romance thrown in. Although one of the main characters is a witch, the story reads a lot like SF.
And here is my list of excellent and original fantasy written from the late 1960's on (some of this is YA):
Ursula K. LeGuin -- "The Earthsea Trilogy"
Patricia McKillip -- Anything by McKillip, but especially the Riddlemaster books, and her recent "Ombria in Shadow."
C. J. Cherryh -- "Fortress in the Eye of Time," "Fortress of Eagles," "Fortress of Owls," and "Fortress of Dragons." Also, "The Goblin Mirror"
Peter S. Beagle -- "The Last Unicorn," "A Fine and Private Place"
Joy Chant -- "Red Moon, Black Mountain"
Marion Campbell -- "The Dark Twin"
Monica Furlong -- "Wise Child"
Alan Garner -- "The Weirdstone of Brisingamen," and "The Moon of Gomrath. Also, "The Owl Service"
Meredith Ann Pierce -- "The Darkangel Trilogy," strictly speaking this would be science fantasy.
Gillian Bradshaw -- "Hawk of May, "The Kingdom of Summer," and "In Winter's Shadow."
Michael Scott Rohan -- "Anvil of Ice," "The Forge in the Forest," "Hammer of the Sun."
Tanith Lee -- Lee writes everything from YA to almost pornographic adult Dark Fantasy. Her best fantasy, in my opinion, falls halfway in between. Would especially recommend "Faces Under Water," "Saint Fire," and "A Bed of Earth: The Gravedigger's Tale."
Paul Hazel -- "Yearwood," and "Undersea."
Ann Downer -- "The Spellkey Trilogy."
Susan Cooper -- "The Dark is Rising," "Greenwitch," "The Grey King," and "Silver on the Tree."
Jack Vance -- "Lyonesse," "The Green Pearl," and "Madouc."
Tad Williams -- "The Dragonbone Chair," "Stone of Farewell," and "To Green Angel Tower."
Kate Elliott -- Crown of Stars series
Katharine Kerr -- Deverry series