Oh my, interesting topic indeed. I might be repeating what others have answered, but since currently I'm reading the sixth book (again, and this time in English!), I feel encouraged to join.
Sauron wasn't killed: since he's a Maiar he cannot die (not even after the destruction of the Ring). At the very least, the Elves did know this, and the Númenóreans should have also, for he was who tricked them and caused them to be the Atalantë. The Valar also knew he was returning, that's why they sent the five Istari: to guide and assist the peoples of Middle-Earth when he returned.
As others have stated, Gondor (and Arnor!) did set watch upon Mordor. Minas Ithil (later Morgul), the Tower of Cirith Ungol (where the orcs take Frodo after he's stung by Shelob), and two towers in the pass between the mountain ranges of Ephel Duath y Ered Lithui (where the black gate is located) were built by the westerners to watch over Mordor. During, like 1500 years or so, Gondor did prevent orcs and haradrim and other foul creatures from returning to Mordor. However, a set of events happened during all this time. Consider the following:
- Isildur gets killed by orcs in the Anduin.
- After Isildur's death, his only surviving son took ownership of Arnor, but didn't claim Gondor, which was instead ruled by the heirs of Isildur's brother and co-ruler, Anarion. Thus the two kingdoms were separated.
- Over time, the kings became obsessed with the lost of longevity. Over the generations, the life length passed from 400 years to 200 and less. Thus the kings dedicated time and resources to find a "cure" to long age (other than being close to Aman).
- Gondor suffered several attacks over time from several fronts: the Harad in the south and the Corsairs of Umbar in the south west, and eventually lost lots of lands there; and also from the Easterlings.
- A civil war ocurred that led to the devastation of Osgiliath and lots of people death.
- The Nazgûl created and built the northern kingdom of Angmar and constantly attacked the norther lands of Arnor, Eriador and Cardolan.
- A great plague struck Middle Earth (even killing the White Tree that Isildur had planted in memory of Anarion).
- The easterlings invaded and devastated Gondor.
- Eventually, Arnor failed and was reduced to nothing, except for the group of rangers known as the Dúnedain.
- The line of Anarion also failed, making way for the House of Stewards.
So, after the war with Sauron, Gondor/Arnor were the only beacon of light for humans in Middle Earth. After they failed, everything else decayed into some sort of Middle Ages, so to speak. It is only natural that they lost the track of things. After all, one is first concerned to bring food to the table or repel the attack from the Corsairs, before sending scouts to see if Barad-dur has been rebuilt.
However, despite this, much was done to prevent evil from returning.
- A White Council was created to counter the ever increasing eveil returning to Mordor. Saruman was tasked to learn about Sauron and prepare a defense. This happened around the same time when Déagol/Sméagol found the Ring (500+ years before the War of the Ring, and 1000+ years after the decline of Gondor started (usually considered at circa 1500 of the Third Age).
- For a long time, Arnor, Rivendell and even Mirkwood kept at bay the attacks of Angmar. Eventually, however, Angmar destroyed the last kingdom of the Dúnedain (Fornost) and Cardolan (where the Barrow-wights were sent, and from where Merry would pick a blade that eventually killed the King Witch of Angmar during the Battle of Pelennor Fields).
- The Beorlings managed to keep the pass between the Misty Mountains and Mirkwood open and orc-free for the most part.
- Rivendell and Lórien kept the orcs of Moria at bay.
- Circa 2000 of the Third Age, the last king of Gondor defeated Angmar.
- During the first 1000 years of the Third Age, or so, Gondor fought and expelled the Easterlings from Rhovanion, Mordor and the Sea of Rhûn.
- Gandalf, the chief strategist, helped the dwarves of Erebor to reclaim their kingdom, and set in motion events that would lead to the death of Smaug and the resurgence of both the dwarves of Erebor and the Iron Hills, and the Kingdom of Dale, eventually getting rid of the orcs from the north (Angmar) and Dol Guldur, which was a serious blow to Sauron (albeit he later sent easterlings to fight Dale, Mirkwood and Erebor during the War of the Ring).
- The White Council eventually drove Sauron from Mirwood. This is important. Although they were late (because of Saruman), this prevented Sauron from learning the whereabouts of the Ring.
- The Dúnedain managed to keep the borders of the Shire well protected, and managed to keep out orcs from the Barrow-wights.
- The elves managed to keep their kingdoms alive due to the power of the three rings. This is important: otherwise, the elves would have had abandoned Middle Earth for Aman long before, and maybe Aragorn wouldn't have been protected, Narsil wouldn't have been restored, Galadriel would have helped The Company, etc etc.
It's just, it wasn't enough. The westerners had to fight not only orcs, but also wicked men and Nazgûls, while at the same time watching how their kingdoms fell apart. It's not as if they forgot to fight or watch Mordor: they were overrun. Add to that Saruman's betrayal (they could have fought Sauron in Dol Goldur 100+ years before the march of Thorin Oakenshield!) and one can see why Sauron gained its former strength.