Tolkien himself covers this quite bluntly in Letter 165. I will try not to say to much and let the quotes do the talking, however we will later see the details are more greatly fleshed out in the same letter and others
The only criticism that annoyed me was one that it 'contained no religion' (and 'no Women', but that does not matter, and is not true anyway). It is a monotheistic world of 'natural theology'. The odd fact that there are no churches, temples, or religious rites and ceremonies, is simply part of the historical climate depicted. It will be sufficiently explained, if (as now seems likely) the Silmarillion and other legends of the First and Second Ages are published.
Letter 165, To Robert Murray, SJ. (draft)
Tolkien goes on to explain, in the same letter, that the men we encounter are those that have been saved from "false worship" and the grasp of the main Evil. Tolkien described it as an "escape" from religion:
Men have 'fallen' – any legends put in the form of supposed ancient history of this actual world of ours must accept that – but the peoples of the West, the good side are Re-formed. That is they are the descendants of Men that tried to repent and fled Westward from the domination of the Prime Dark Lord, and his false worship, and by contrast with the Elves renewed (and enlarged) their knowledge of the truth and the nature of the World. They thus escaped from 'religion' in a pagan sense, into a pure monotheist world, in which all things and beings and powers that might seem worshipful were not to be worshipped, not even the gods (the Valar), being only creatures of the One. And He was immensely remote.
The High Elves were exiles from the Blessed Realm of the Gods (after their own particular Elvish fall) and they had no 'religion' (or religious practices, rather) for those had been in the hands of the gods, praising and adoring Eru 'the One', Ilúvatar the Father of All on the Mt. of Aman.
The Númenóreans thus began a great new good, and as monotheists; but like the Jews (only more so) with only one physical centre of 'worship': the summit of the mountain Meneltarma 'Pillar of Heaven' – literally, for they did not conceive of the sky as a divine residence – in the centre of Númenor; but it had no building and no temple, as all such things had evil associations. But they 'fell' again – because of a Ban or prohibition, inevitably.
[Sauron] steadily got Arpharazôn's mind under his own control, and in the event corrupted many of the Númenóreans, destroyed the conception of Eru, now represented as a mere figment of the Valar or Lords of the West (a fictitious sanction to which they appealed if anyone questioned their rulings), and substituted a Satanist religion with a large temple, the worship of the dispossessed eldest of the Valar (the rebellious Dark Lord of the First Age).
Tolkien illustrates the idea that in Númenor they had a place to invoke the name of Eru, and admire or remember him, but in time they were corrupted and fell to evil. Beginning to worship Melkor, through the deceits of Sauron.
The descendants of the Númenoreans without the Meneltarma as a place of worship led to a refusal to worship. The Kings of Númenor, however, were still able to visit the Mindolluin and there offer thanks to Eru and of a form repeat what their ancestors had done on Meneltarma
But the 'hallow' of God and the Mountain had perished, and there was no real substitute. Also when the 'Kings' came to an end there was no equivalent to a 'priesthood': the two being identical in Númenórean ideas. So while God (Eru) was a datum of good Númenórean philosophy, and a prime fact in their conception of history. He had at the time of the War of the Ring no worship and no hallowed place. And that kind of negative truth was characteristic of the West, and all the area under Numenorean influence: the refusal to worship any 'creature', and above all no 'dark lord' or satanic demon, Sauron, or any other, was almost as far as they got. They had (I imagine) no petitionary prayers to God ; but preserved the vestige of thanksgiving. (Those under special Elvish influence might call on the angelic powers for help in immediate peril or fear of evil enemies.) It later appears that there had been a 'hallow' on Mindolluin, only approachable by the King, where he had anciently offered thanks and praise on behalf of his people; but it had been forgotten. It was re-entered by Aragorn, and there he found a sapling of the White Tree, and replanted it in the Court of the Fountain. It is to be presumed that with the reemergence of the lineal priest kings (of whom Lúthien the Blessed Elf-maiden was a foremother) the worship of God would be renewed, and His Name (or title) be again more often heard. But there would be no temple of the True God while Númenórean influence lasted.
But if you imagine people in such a mythical state, in which Evil is largely incarnate, and in which physical resistance to it is a major act of loyalty to God, I think you would have the 'good people' in just such a state: concentrated on the negative: the resistance to the false, while 'truth' remained more historical and philosophical than religious.
Finally, in Letter 142, an earlier letter to the above draft, Tolkien famously admits that is work is indeed religious, however due to the fantastical nature of the story, has removed the elements of religion as is known to us (Letter 131)
The Lord of the Rings is of course a fundamentally religious and Catholic work; unconsciously so at first, but consciously in the revision. That is why I have not put in, or have cut out, practically all references to anything like 'religion', to cults or practices, in the imaginary world. For the religious element is absorbed into the story and the symbolism.
Letter 142, To Robert Murray, SJ.
Further information can be found in the famous Letter to Milton Waldman, 131, and Letter 153, a draft to Peter Hastings. More in depth discussion can also be found in Purthill's Tolkien: Myth, Morality and Religion