Well, I don't know what this says about me, but my two favorite Tolkien romances both end badly.
I am fond of Aldarion and Erendis. Their story is so rich and so human.
I also really like the romance between Eowyn and Aragorn. (*cringes and waits for the Faramir brigade to descend from the ceiling*) Even though it happens largely in Eowyn's mind, and even though she later professes that she loved "a shadow and a thought," her love is real at the time she feels it. And who can blame her? The only men she knows of her own rank are her brother and her uncle; her only "suitor" is the despicable Grima; and suddenly this man shows up who is the embodiment of all she's ever wanted. He represents escape, strength (in opposition to her downtrodden uncle), loyalty, and hope. Of course she attaches all of her hopes to him, and that must include love. Tolkien's characters never fall in love willy-nilly; their marriages and even their unrequited desires always serve a purpose. Aragorn must marry Arwen because she complements his lineage and his importance. Rosie must wait for Sam to reinforce the homely virtues of the Shire. Likewise, Eowyn must fall in love with Aragorn. It makes real her spirit and her anguish, as well as pointing out (as the story does again and again) that he is irresistible--not only to her as a woman, but to everyone he encounters, as a leader. Everyone looks to him.