Having been rather obsessed by George R. R. Martin's
A Song of Ice and Fire series recently
I feel like adding my two cents. Martin definitely gives a totally different feel than Tolkien, yet in some ways he is closer to Tolkien than most contemporary fantasy writers. It is because of the depth and the scale of his creation, and how the world feels so very real. True, he doesn't have Tolkien's skill and interest in linguistics (which is something he seems to like to repeat in interviews) or mythological depth, but I find his cities, families, people, institutions, society and religion exceptionally well thought of. Like
Ang and
Nog have said, he does do some stuff better than Tolkien - if we can compare them, after all, their styles are completely different although both fall under the genre of fantasy.
As for who creates a similar feeling as Tolkien, I have two names in mind although for very different reasons. Ursula Le Guin, a genius in her own right, is the only one who can rival (and don't kill me, maybe surpass) Tolkien in writing grand-scale bittersweet mythological fantasy that you simply cannot forget and have to love. Like Tolkien, she has the gift of combining legendary and philosophical stuff with a good plot and interesting characters and balancing with it neatly without making it too shallow or alternatively pompously boring. Suffice to say, I admire her greatly.
The one who can reach any kind of Tolkieny feeling in a very different way is Guy Gavriel Kay. He helped Christopher Tolkien with constructing the Silmarillion and you can see how deep he is in the Tolkienian mythology if you read his earliest novels,
The Fionavar Tapestry trilogy. He doesn't really add anything new though, just recycles some of the coolest ideas from the Silmarillion alongside with more epic fantasy stuff and even King Arthur. I think
The Fionavar Tapestry is slightly over the top and it would be very easy to criticise it, yet it still is a very good series, mostly thanks to Kay's touchingly epic writing style and his beautiful language. (I have to say though that I prefer his latter novels.) Those two things are actually why Kay is in a way closest to Tolkien in the contemporary fantasist: his writing is sometimes like reading a poem, and when he writes epic, it is very epic ŕ la Rohan had come at last.