The following is adapted from Jeff "Colonel Hardisson" Black's October 2007 blog post "Some thoughts on the running of a Fourth Age Middle-earth game campaign".
A Fourth Age Middle-earth campaign is a great way to use Tolkien's world as a game setting without having to worry about, or change, canon. Tolkien himself provides a lot of interesting detail about his world after the time of the Lord of the Rings, detail that provides a wealth of campaign hooks and ideas. I'll quote some of this material to show how relevant it is to a gamemaster wanting to run a campaign in Middle-earth.
"For though Sauron had passed, the hatreds and evils that he bred had not died, and the King of the West had many enemies to subdue before the White Tree could grow in peace. And wherever King Elessar went with war King Eomer went with him; and beyond the Sea of Rhun and on the far fields of the South the thunder of the cavalry of the Mark was heard, and the White Horse upon Green flew in many winds until Eomer grew old."- The Return of the King, Apendix A, part II, The House of Eorl
This paragraph sets the premise for decades of Fourth Age adventure. It doesn't involve just Men, either. Even though Tolkien makes much of how the Fourth Age is a time of fading for all the peoples of Middle-earth except Men, it is apparent that many of the other races are actually becoming much more gregarious in the Fourth Age. Here are some examples:
"After the fall of Sauron, Gimli brought south a part of the Dwarf-folk of Erebor, and he became Lord of the Glittering Caves. He and his people did great works in Gondor and Rohan. For Minas Tirith they forged gates of mithril and steel to replace those broken by the Witch-king. Legolas his friend brought south Elves out of Greenwood, and they dwelt in Ithilien, and it became once again the fairest country in all the westlands."- The Return of the King, Appendix A, part III, Durin's Folk
"Three times Lorien had been assailed from Dol Guldur, but besides the valour of the elven people of that land, the power that dwelt there was too great for any to overcome, unless Sauron had come there himself. Though grievous harm was done to the fair woods on the borders, the assaults were driven back; and when the Shadow had passed, Celeborn came forth and led the host of Lorien over Anduin in many boats. They took Dol Guldur, and Galadriel threw down its walls and laid bare its pits, and the forest was cleansed.
In the North also there had been war and evil. The realm of Thranduil was invaded, and there was long battle under the trees and great ruin of fire; but in the end Thranduil had the victory. And on the day of the New Year of the Elves, Celeborn and Thranduil met in the midst of the forest; and they renamed Mirkwood Eryn Lasgalen, The Wood of Greenleaves. Thranduil took all the northern region as far as the mountains that rise in the forest for his realm; and Celeborn took all the southern wood below the Narrows, and named it East Lorien; all the wide forest between was given to the Beornings and the Woodmen. But after the passing of Galadriel in a few years Celeborn grew weary of his realm and went to Imladris to dwell with the sons of Elrond. In the Greenwood the Silvan Elves remained untroubled, but in Lorien there lingered sadly only a few of its former people, and there was no longer light or song in Caras Galadon."- The Return of the King, Appendix B, The Tale of Years
In addition, it seems that the Fourth Age is a time of renewal and rebuilding. A few brief but significant passages suggest that Elessar seeks to reinvigorate the entire northwestern part of Middle-earth, something which will take a number of hardy individuals performing countless heroic deeds. Here are some examples:
"King Elessar rides north, and dwells for a while by Lake Evendim." [Fourth Age 14]- The Return of the King, Appendix B, The Tale of Years
"And Aragorn gave to Faramir Ithilien to be his princedom, and bade him dwell in the hills of Emyn Arnen within sight of the City.'For,' said he, 'Minas Ithil in Morgul Vale shall be utterly destroyed, and though it may in time to come be made clean, no man may dwell there for many long years.'"- The Return of the King, Book VI, Chapter V: The Steward and the King
As simple as these passages seem, the actual doing of the deeds required to accomplish them is rather difficult. Elessar not only travels to, but actually lives at Annuminas for a while. Seeing that the city was a ruin at the end of the Third Age, abandoned for centuries, and surrounded by wilderness, this implies an enormous effort was made by Gondor. It seems clear that Annuminas is restored, and becomes the chief city of the northern kingdom. It is not difficult to picture that King Elessar also rebuilt Fornost Erain and Tharbad. It is also highly likely, even though unmentioned, that Osgiliath was cleared and rebuilt; given Elessar's desire to rebuild the Dunedain kingdoms, this seems a natural assumption to make.
In addition, the destruction of Minas Ithil, an entire fortress-city the size of Minas Tirith (at least), held by the chief of the Nazgul for centuries and used as a base for a large part of the strength of Mordor, is a task which would have to involve thousands of troops. This is especially true if the city was used as a rallying point and base for some of the remnants of Sauron's armies. It is not hard to imagine that the interior of the city is filled with all manner of evil things - Men, beasts, traps, and a generally unwholesome atmosphere. Very much like a D&D adventure. And speaking of D&D adventures...
"Far, far below the deepest delvings of the Dwarves, the world is gnawed by nameless things. Even Sauron knows them not. They are older than he. Now I have walked there, but I will bring no report to darken the light of day."- The Two Towers, Book III, Chapter V: The White Rider
With the Balrog defeated, Sauron gone, and a huge part of the strength of the orcs in the region destroyed, it is quite likely that the Dwarves would at least begin to explore and clean out Moria. This would provide the archetypal dungeon adventure. Especially given that tantalizing glimpse Gandalf gives into a world of darkness and horror far beneath even where the Kings of the Dwarves once dwelled.
Here are a few random, and final, thoughts about possibilities for Fourth Age campaigns:
Mirkwood (renamed Eryn Lasgalen) and Ithilien would likely also remain havens for a time for evil creatures, such as spiders and orcs, until the Elves, woodmen, and Beornings finally rid the forest of them. Shelob still dwells near Cirth Ungol, and the Watcher in the Water still haunts the lake near the west gate of Moria.
Veterans of Celeborn's taking of Dol Guldur would be good candidates for duty in Mordor itself, or any place wherein darkness still holds sway. After Sauron's defeat at the end of the Second Age, Gondor built fortresses and watchtowers in Mordor itself - in fact, the great gates of Mordor, Cirith Ungol, and Minas Morgul were all built by the Kings of Gondor to keep watch on Mordor and ensure Sauron couldn't return. There is no reason to assume that the new king would not also build fortresses to provide a forward military presence in Mordor. Sauron is gone, and so are his most powerful lieutenants, but there were certainly other figures of great power who would like to make Mordor their base.
There are at least two great Elves still unaccounted for in the 4th Age: Daeron and Maglor. Perhaps one or the other is encountered in the East, or perhaps one or the other returns to western Middle-earth, finally weary of wandering, their pain and sorrow driving them towards the west at last. Perhaps one of them could take up residence in fading Lorien, or even in Rivendell for a time, or take up the kingship of East Lorien once Celeborn leaves. In addition, the only other wizard named in the Lord of the Rings books and unaccounted for at the end of the story, Radagast the Brown, is also still present, as are the two mysterious "Blue Wizards" - mentioned in notes and essays Tolkien wrote and never published in his lifetime, but which can be found in books like Unfinished Tales - who went far into the East and were never heard from again.
All in all, Middle-earth of the Fourth Age is a vital, interesting setting. The gloom of Sauron's presence has been replaced by an atmosphere of beginnings, but remnants of the Shadow and the things it wrought are still in Middle-earth, awaiting heroes to root them out and vanquish them. Just because the One Ring and its master are gone doesn't mean their legacy has vanished.