I think Tolkien certainly had an influence on D&D, as did many other fantasy authors, as well as other sources. I think that Tolkien's influence on D&D becomes a talking point because of early rebuttals by Gygax (and legal issues, IIRC—FWIW, I believe there has been mention/discussion of the Tolkien estate suing over use of IP, and thus why halflings are halflings instead of hobbits, treants aren't ents, balors aren't balrogs, etc.).
The elements are present there, but changed just enough to legally be different (sort of like Adidus shoes or Levvy Jeans). The big thing, for me at least, is the use of mithral in D&D and orcs. The original mithril in Tolkien's works is memorable, and it sees use in other settings with a vowel change.
(For those who don't know, mithril, like balrog, both mean something in Tolkien's created languages [grey brilliance and powerful demon respectively, IIRC].)
Orcs can be linked back to sources earlier than Tolkien's, but it is the goblin-like orc of Tolkien's works that serves as the wellspring for the orcs or today.
The style of the initial demihumans (to use the AD&D terms) was very Tolkienesque, as well. The halfling subraces of Stout, Hairfoots, and Tallfellow echoed Tolkien's Stoor, Harfoot, and Fallohide hobbit stock. The core "high" elves were much like Elrond's folk, with the "grey" elves mirroring Galadriel's Galadrim, and the "wood" elves mirroring the elves of Mirkwood. The hill and mountain dwarves mirror Dain's dwarves from the Iron Hills and Thorin's 13 from the Lonely Mountain.
However, the influences of other authors on the game are just as significant, if not moreso. Vancian magic is a fine example, as are several others mentioned in previous posts. However, the prominence of these influences had faded (to a degree) with each reworking of the game.