This discussion was had on here not that long ago—my opinion was, and still is, that she took way more from Tolkien than people are actually suggesting. The Dementors and Nazgûl aren't only similar visually: read TLOTR and marvel at how precisely how similar the effect the Nazgûl has on the hobbits is to the effect of the Dementors. The way it's written is extraordinarily similar. Even more outrageous is the (I believe single) use of "he who we do not name" (or something along those lines) for Sauron, which then became the moniker for Voldemort.
Thing is, I don't think that takes away from Rowling's act of genius, which was mapping existing fantasy tropes and elements found specifically in TLOTR onto a modern child (ie her readership). But to suggest that she didn't take anything from Tolkien is also to suggest that if Tolkien didn't exist her books would have been identical. Which is completely untrue.