[Amarie of the Vanyar]

With regards to the what happens to Men after death, Aragorn says to Arwen:

"But let us not be overthrown at the final test, who of old renounced the Shadow and the Ring. In sorrow we must go, but not in despair. Behold! We are not bound for ever to the circles of the world, and beyond them is more than memory, Farewell!"
Tale of Aragorn and Arwen; Appendix A

Although he doesn't explains in detail what awaits them after death, there is the promise of something; something like a 'new life'.

And in the Athrabeth Finrod ah Andreth, Finrod guesses that Men don't belong to Marred Arda:

'who is but a guest here in Arda and not here at home, whith a House [hröa] that is built of the matter of Arda and must therefore (one would suppose) here remain? [...]
Then "death" would (as I said) have sounded otherwise to you: as a release, or return, nay! as going home!'
Athrabeth Finrod ah Andreth; HoME 10

And as they dialogue goes on, he concludes that the men's fëa will 'build' a healed hröa:

'Then this must surely follow: the fëa when it departs must take with it the hröa. And what can this mean unless it be that the fëa shall have the power to uplift the hröa, as its eternal spouse and companion, into an endurance everlasting beyond Ëa and beyond Time?
Athrabeth Finrod ah Andreth; HoME 10

And after that, Finrod has a vision of Arda Healed:

'And then suddenly I beheld as a vision Arda Remade; and there the Eldar completed but not ended could abide in the present for ever, and there walk, maybe, with the Children of Men, their deliverers, and ding to them such songs as, even in the Bliss beyond bliss, should make the green valleys ring and the everlasting mountain-tops to throb like harps'
Athrabeth Finrod ah Andreth; HoME 10