In Tolkien's lore, they are many very old beings. Is there any indication that some or even one would have been truly immortal and survived into what we would call "our" history or Age? Galadriel, Cirdan, Ingwe, Bombadil, Ungoliant, or any Maia or Ainur? Anyone or anything? Seems there must be some kind of carry-over seeing as how "man" existed in this time.
Tolkien does better than hint. In the beginning of The Hobbit he states that hobbits may continue to exist now and that we don't notice because they keep out of our way. This is from the opening page:
I suppose hobbits need some description nowadays, since they have become rare and shy of the Big People, as they call us. They are (or were) a little people, about half our height, and smaller than the bearded Dwarves.
He may well have gone back on this suggestion at some point, but it might be worth looking out for them next time you're in the English countryside. You'll have to be quiet though because:
There is little or no magic about them, except the ordinary everyday sort which helps them to disappear quietly and quickly when large stupid folk like you and me come blundering along, making a noise like elephants.
All the immortal beings - elves, Ainu, etc - are still around.
Tolkien is quite clear that the Elves at least are "bound to the circles of the World" and will endure as long as it does. They may not be present in Middle-Earth itself, but they are certainly still alive somewhere.
The Vala and Maia also, when they descended into Arda, committed themselves to existing for as long as it does, and cannot leave before the end.
Eru Ilúvatar: He is The One, he was since ever and will be forever, but he is not "here" on Earth.
The Ainur (not Valar/Maiar): They are with Eru, also out of this world.
The Valar and the Maiar: They are eternal as well, and they are on Arda until the end of the world, but they are on Valinor, which is now removed from the world as we know it.
Elves: Those who were on Valinor continue there as the Valar and Maiar. Those who were on Middle Earth (either after return (Noldor) or those who never went to Valinor) left Middle Earth through the "straight path" towards Valinor. They'll be there, either live or dead on Mandos' "forever hotel", until the end of the world.
Dwarfs, hobbits: not immortals. Species may be still around, but no individuals. Ringbearers, however, are in Valinor.
Men: not immortals, by definition. Well, Duncan McLeod...
Orcs: Let's hope they were all killed.
Ents: It is unknown if they are truly immortal. If they are like trees, they are not, but they live way long. Some of the forests in Central Europe may have Treebeard still around.
Tom Bombadil: As we do not know who he really is, we can not know. But it is a good bet to say he is still in his small realm, probably around central France.
Note: Central Europe and Central France locations move a lot depending on the equivalences you do between northwestern Middle Earth and modern Europe.
EDIT: I forget about Arien and Tillion, who drive the Sun and the Moon, respectively. They are undoubtedly still performing their work.
To add to the excellent previous answers, Galadriel herself says that some of the Elves may have lingered in Middle-earth, even into our own times.
"Yet if you succeed, then our power is diminished, and Lothlórien will fade, and the tides of Time will sweep it away. We must depart into the West, or dwindle to a rustic folk of dell and cave, slowly to forget and to be forgotten."
-The Lord of the Rings, The Fellowship of the Ring, Book II, Chapter 7: "The Mirror of Galadriel"
After the destruction of the One Ring, the power of the Three Rings of the Elves would also end and the Age of Men would begin. Elves that remained in Middle-earth were doomed to a slow decline until, in the words of Galadriel, they faded and became a "rustic folk of dell and cave," and were greatly diminished from their ancient power and nobility. While the power of the remaining Noldor would be immediately lessened, the "fading" of all Elvenkind was a phenomenon that would play out over hundreds and even thousands of years; until, in fact, our own times, when occasional glimpses of rustic Elves would fuel our folktales and fantasies.