In the Silm, there are fleeting references to the Numenoreans who went to Meneltarma at the 'appointed seasons' to worship Iluvatar, and it is also said that the Faithful 'kept Iluvatar in their hearts.' These particular references are from the Akallabeth. Other than that, and what Sharku has pointed out, I have no instances that I can recall from memory from the other books of any sort of worship rituals or holy peoples.
Perhaps religion was left out of Tolkien's works deliberately because he did not want to start a cult, as Sharku's quote alludes to. Being a devout Catholic, I doubt that Tolkien would have wanted to be responsible for drawing people away from what he considered the true way by adding details of the religious practices of ME.
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'Dulaman na Binne Bui, Dulaman Gaelach/ Dulaman na farraige, 's e b'fhearr a bhi in Eirinn!'
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