There are some explicit statements in Letters and LotR itself. Letter 144 notes that Dragons survived and were active close to our time, whereas LotR is conceived as being drawn from a surviving copy of the Red Book of Westmarch.
From the History of Middle-earth, it's definitely the case that Ambarkanta Map V suggests Africa, and the British Isles were originally thought of as being the remnants of Beleriand after the War of Wrath (a concept that was later abandoned).
The LotR prologue (Concerning Hobbits) also claims that Hobbits are still around:
Hobbits are an unobtrusive but very ancient people, more numerous formerly than they are
Otherwise, what Tolkien wrote about was a world in a perpetual state of decay, and the passing from mythical times to historical times. Gandalf's words to Aragorn on Mindolluin summarize it best:
This is your realm, and the heart of the greater realm that shall be. The Third Age of the world is ended, and the new age is begun; and it is your task to order its beginning and to preserve what may be preserved. For though much has been saved, much must now pass away; and the power of the Three Rings also is ended. And all the lands that you see, and those that lie round about them, shall be dwellings of Men. For the time comes of the Dominion of Men, and the Elder Kindred shall fade or depart.
It's clear that even the Renewal at the end of the Third Age was not accomplished without grievous loss, and this is all wrapped up in a central theme (perhaps the central theme) of Tolkien's: that of the Marring of Arda and the Falls of Men and Elves.
That's all something that would take a full book to discuss in more detail, but fortunately for you that book has been published and it's called History of Middle-earth 10: Morgoth's Ring. I'll refer you to that if you're interested in more information about this theme.