Despite the fact that it often seems like we're whirling through a timeless landscape, dates are actually quite important to the overall structure of The Lord of the Rings series. The fact that Frodo starts off his quest in the fall—when things start to get cold and miserable—and Aragorn takes his throne in the spring—when the world is reborn again—demonstrates the symbolic significance of seasons to convey tone in Tolkien's work. Frodo starts his dark quest as the world is literally shutting down for the year, while Aragorn begins to restore Gondor just as the season is getting warmer and more welcoming.
Beyond these images of the seasons, Tolkien also bothers to stop and tell us that Frodo's injury at Weathertop in The Fellowship of the Ring happens on October 6th. In The Return of the King Frodo is taken captive and held at Cirith Ungol on March 13th, and Sauron falls on March 25th. Thus, March 25th becomes the date of the Gondorian New Year. Arwen arrives in Minas Tirith the day before Midsummer—probably around June 20th—and she and Aragorn marry on June 21st, the day of the summer solstice, the longest day of the year.
At first, it may just seem like a note of passing interest that Tolkien specifies these dates. He's showing us that the whole action of The Lord of the Rings, from Frodo's departure from the Shire to his return, takes just over one year (which is amazing when you think about just how much everything has changed).
But then, Frodo gets sick on the exact anniversaries of his injuries: his illnesses come on March 13th (Shelob) and on October 6th (the Wraith knife). The predictable return of Frodo's sickness proves that something important is going on with the calendar in these novels.
The thing about dates is, they allow us to mark the progress of time forward. But dates and seasons both repeat themselves in a regular cycle. Even if the characters of The Return of the King have defeated Sauron, they can't prevent winter—or evil itself—from returning to the world. Just look at Sharkey rearing his ugly head.
In any case, even though Frodo has done the task set for him, he can't totally move on from his experiences in Mordor. The anniversaries of his injuries still come back to hurt him. So there is an overall sense of progress in this novel, since Sauron is defeated forever. But there is also the lingering feeling that defeating evil cannot be permanent. The people of Middle-earth still remember the evils done in Sauron's name. And they had better watch out for lingering evil around them.