Imagine no religion

[Inziladun]

Eru Ilúvatar is the Prime Creator in the mythos. I think it can be said the Elves and Dúnedain at least worshipped him. The Elves, at least the Noldor and Sindar we see, were followers of the Valar, who were the 'governers' appointed by Eru, so they worshipped him by proxy. The Avari probably did too, since by the time of the Second Age, at least, they were mostly led by Sindarin rulers anyway.
The Dúnedain in Númenor worshipped Eru directly on the Meneltarma, as far as I know the only people to ever do so. It seems to me they did so because, unlike the Eldar, they had little to do with the Valar, who were the intermediaries between Eru and his Children. That tradition apparently was not continued in their Realms In Exile, but they had not forgotten the Valar, or the One. There is a mention of Dúnedain appealing to the Valar in LOTR. Mablung, or Damrod, I don't recall, said this when the oliphaunts appeared in Ithilien:

'Ware, ware, the Valar turn him aside!'

And what of the Oath of Cirion to Eorl the Young at Elendil's tomb?

'This oath shall stand in memory of the glory of the land of the Star, and of the faith of Elendil the Faithful, in the keeping of those who sit upon the thrones of the West and of the One who is above all thrones forever.'

The Dwarves would have given their attention to Aulë, again another proxy for Eru.

As to whether there was any organised worship, there doesn't appear to be outside the practices of the Númenóreans.

The Shire-folk make no reference to knowledge of the Valar or the One, and I don't know if that speaks to mere ignorance, as they had become ignorant of so many things in Middle-earth, or something else.

x/d with The Mouth of Sauron, who noted the same quote from one of Faramir's men