I know that The Lord of the Rings is located in Middle-earth and by the looks of clothes and props, we're looking at well before the 18th century.
What's the best guess at the time period that The Lord of the Rings was set in?
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About 6000 years ago. But I don't think the exact details of when were at all important to Tolkien.
Tolkien had written multiple times that Middle-earth is located on our Earth. He has described it as an imaginary period in earth's past, not only in The Lord of the Rings (see Prologue and Appendices), but also in several correspondence letters, estimating the end of the Third Age to about 6,000 years before his own time, and in N.W. Europe (Hobbiton for example was set in same latitude as Oxford), though at times he would also describe elements of the stories as a kind of "...secondary or sub-creational reality" or "Secondary belief" in replies to letters.
It was meant to be in this world, about 6,000 years ago. In Letter #211 Tolkien addressed this specifically (the asterisk is to the footnote, reproduced below):
I hope the, evidently long but undefined, gap* in time between the Fall of Barad-dur and our Days is sufficient for 'litereary credibility', even for readers acquainted with what is known or surmised of 'pre-history'.
*I imagine the gap to be about 6000 years: that is we are now at the end of the Fifth Age, if the Ages were of about the same length as S.A. and T.A. But they have, I think quickened; and I imagine we are actually at the end of the Sixth Age, or in the Seventh.
I'm pretty sure that Tolkien intended his work to be a mythology of our ancient past. It has been speculated that if LOTR ends at the beginning of the 4th age, then we may currently be in the 7th age. That said, the length of an age is arbitrary and the events of the 3rd age do not fall anywhere in our own history. You could think of the 7th age as all of known human history. Anything older than that can only be considered myth.
Edit: please note that the question explicitly states the time frame from 'the real world' and makes references to film props etc. therefore what the author intended is not the question. The questions is purely concerned with where in the course of real history the films would be fit in.
If you wish to downvote, please feel free but could you also leave a comment so I know why?
Looking at the films Gondor is about 1450 and the shire is about 18c.
Please note this is based upon the full plate armour used by gondor's soldiers, orcs and elves (eg barbutes), the pole block tactics used by the orcs, the agricultural machines used in the shire and the crockery and cutlery used by hobbits etc.
In 1960, Tolkien said that the year "Bel. 310" was 16,000 years ago.
If they [Men] awoke in VY [=Valian Year] 1050 that would give 40 VYs, or 5,760 Sun Years in which Melkor could have dealings with them and corrupt them, before his captivity. The Atani entered Beleriand in 310 Bel. That is in the 22nd Sun-year of VY 1498. Men had then existed for 448 VYs + 22 SYs: i.e., 64,534 Sun Years, which, though doubtless insufficient scientifically (since that is only – we being in 1960 of the 7th Age – 16,000 years ago: total about 80,000), is adequate for purposes of the Silmarillion, etc.
The Nature of Middle-earth - "The Awaking of the Quendi"
Carl Hofstetter notes that the year "Bel. 310" was 6,752 years before the end of the third age.
Therefore, if Men entered Beleriand in Bel. 310, and the First Age ended c. Bel. 600 (cf. XI:346), then that entrance occurred 290 + SA 3441 + TA 3021 = 6,752 years before the end of the Third Age.
The Nature of Middle-earth - "The Awaking of the Quendi", note 30
Adding on another 61 years brings us from 1960 to 2021.
16,000 - 6,752 + 61 = 9,309
Tolkien's 16,000 year estimate was probably rounded, so 9,300±500.
Note that some of the other answers quote a letter from Tolkien where he gave an estimate of 6,000 years. However that letter was written in 1958, two years prior to the text I've quoted. Also that letter is written in more ad-hoc style, without the precise calculations found in this text.
C.S. Lewis' Space Trilogy and Tolkien's Lord of the Rings Trilogy was a collaboration, taking place in our universe. In fact, C.S. Lewis referred to Numinor several times over the course of his book That Hideous Strength, which was set in post-WWII England.
From the Prelude he writes:
Those who would like to learn further about Numinor and the True West must (alas!) await the publication of much that still exists only in the [manuscript] of my friend, Professor J. R. R. Tolkien.
According to the antagonists in C.S.Lewis' book That Hideous Strength, it would have been sometime before 110,000 years ago (before the beginning of the last glacial period).
In discussing Merlin, Frost states:
"What we have here," said Frost pointing to the sleeper, "is not, you see, something from the fifth century. It is the last vestige, surviving into the fifth century, of something much more remote. Something that comes down from long before the Great Disaster, even before primitive druidism; something that takes us back to Numinor, to pre-glacial periods."
Of course, these are the same folks that predicted that Merlin would join their side, and were subsequently devoured by their own future vivisection experiments; take what they say with a grain of salt. ;)
Seeing how Lord of the Rings stems from Beowulf and the Nibelung saga as well as the Poetic Edda, 5th to 6th century would be a good guess.
The sword that was broken and remade, the cursed ring, invisibility, the dragon and his hoard, and the stolen goblet, and even many of the names (Gandalf, Balin, Durin, anyone?) stem from there.