Well I'll jump in with some recommendations that I haven't seen in previous posts. None of these are exactly like JRRT, and all of them def have a more "contemporary" feel, which is natural because they are all living now, and English has changed since JRRT's time. However, they do share some of the characteristics I love with JRRT...
Robin McKinley: overlaps with JRRT in that there is fantasy, vivid world-creation, and (dare I say it) reading her stories gives me a sense of history behind the story, world and geography beyond the borders of each story, that I think is remeniscent of Tolkien's hints of the "greater" stories behind and beyond and to the sides of the stories he tells.
In particular, for any who haven't ready her work, I'd start with the Damar books (The Hero and the Crown and The Blue Sword) OR Sunshine.
Diane Duane: overlaps with JRRT in the affirmation of life, the plots of good vs. evil, the choice of each character to decide not what times they are born into, but what to do with the times they are given, and (a little bit) in creating phrases and words in invented languages - although JRRT made his plots around his language, and I think DD's may have been more the other way around. She may have more resonances with C.S. Lewis than Tolkien, but then CSL and JRRT have a lot of resonances with each other so...
For any who haven't read her works, I'd start with the Young Wizards series, which is supposedly geared towards young adults but which can be enjoyed by any age. if anyone likes star trek I'd also highly recommend her Rihannsu trek books, which create a whole Romulan culture (not acknowledged by the later tv & movie 'trek canon' which is FASCINATING.
Michelle Sagara West: I happen to be enjoying her "Chronicles of Elantra" series which overlaps with JRRT in creating cultures and non-human races that live and interact with one another, - Human, Barrani (similarities with elves), Dragons, Leontines (basically sentient talking lions who walk upright), Tha'alani, Aerians (people with wings), Norannir, Shadows... and of course, good vs. evil.
Marie Brennan's "Memoirs of Lady Trent" series: overlaps with JRRT in that I get a sense of realism from these books that reminds me of JRRT. Also, the pacing is slower than some other works, which reminds me of JRRT lingering over the introductions of characters and the start of plots in the Hobbit and FotR. The setting's sort of alternate-universe instead of "a long long time ago in OUR timeline", and it's only a bit archaic - but that archaism, though different, reminds me of JRRT. Also, the story framing idea, that this story is being told or written by one of the characters, reminds me of Bilbo and Frodo who supposedly wrote the Hobbit and LotR after going through the events themselves.