Many people seem familiar with Lord of the Rings via the Jackson movies, but there's a much longer tradition of artists depicting scenes from the books. Plus of course, two TV / movie series that bring some great voicework & songs (Rankin-Bass' version) and fascinating animation (Ralph Bakshi's), despite their overall flaws.
But in terms of alternate illustrators, check out the Brothers Hildebrandt, for example.
Personally, I'm enjoying this Argentinian master's version because the characters are no longer as wholesome-looking as in other depictions, giving the scene slight horror overtones. The ammonite is a nice bonus, too!
Although I also really like the Alan Lee/movies art style, having alternative ways to visualize such an immense world is wonderful. The Hildebrandt brothers' illustrations especially have a more medieval, almost naive touch to them that I think lends itself well to fairy tales.