After all, D&D was really an interactive version of Tolkein's world to begin with, wasn't it?
No, though Tolkien's work had some influence and helped create a market for the game.
Gary Gygax was an avid war gamer, a published author of articles on wargames, and a developer of wargaming rules. The D&D game evolved out of wargaming rules for medieval-style battles - knights, archers, that kind of thing.
You can see this even in just the cover description from the early rules: my copy of Greyhawk says "Rules for Fantastic Medieval Wargames Campaigns Playable with Paper and Pencil and Miniature Figures". Similarly, if you read the history of the game you'll find references to the "Castle & Crusade Society", part of the International Federation of Wargaming. I believe GenCon - the convention people now associate with D&D - was originally the Lake Geneva Wargames Convention.
Gygax, a major bookworm, drew upon a wide variety of ideas in comic books and in early science fiction and fantasy (much of which predated Tolkien), as well as mythology (predating Tolkien by thousands of years). For example, I see gargoyles, chimera, griffins, and medusae in one of the early rule books (none of which are to be found in Tolkien).
Many people seem to think that Tolkien created fantasy, but there are many earlier and contemporary works in this genre. For example, the first "Conan the Barbarian" story (Robert Howard) was written in 1932, the characters "Fafhrd and the Gray Mouser" (Fritz Leiber) were created in 1934, and the Hobbit would only be published in 1937.
The key innovations that Gygax (and others) developed from earlier war games were to reduce the army size to focus on a small number of heros, to add fantasy and science fiction elements, and to have the idea of an ongoing game (the "campaign").
Dave Arneson, the other major participant in creating the game, was probably the first to add science fiction elements, as part of his Blackmoor campaign (which I understand grew out of his wargaming sessions). This, of course, was very different from anything in Tolkien.
You can still dig up copies of the old rules and see the wargaming elements (it helps if you're already familiar with old school wargaming). As the game evolved, the wargaming-style rules started to disappear. The use of miniature figures, for example, stopping being important, the wargamers had lots of these, and terrain to go with them, but as the game moved beyond wargaming it started to attract players that didn't have or want any of this stuff.
In today's RPGS, it's been my experience that we tend to get two different styles of play: some groups emphasize the role playing, others emphasize the gaming (rules and tactics). The latter approach is much closer to the original game, but both approaches can make for reasonable and enjoyable play sessions. Many current gamers are firmly wedded to one camp or the other, and this greatly complicates producing new or updated RPG rule sets, as the two groups want fundamentally different things in the rules they use. For example, to the groups that emphasize role playing, complex rules (a long tradition in wargaming) just get in the way, while to the groups that emphasize tactical play, these actually provide more freedom and opportunities for tactical creativity.