#Nothing special
- As mentioned in the previous answer, Grievous collected and used lightsabers of the Jedi he defeated.
- Luke owned his father's lightsaber (given to him by Obi-Wan in ANH). Though Anakin was not dead, he was defeated by Obi-Wan after changing his force allegiance.
- After Anakin Skywalker died (in RotJ), the same lightsaber was then
used by Finn, then Rey in Episode 7 to nearly defeat Kylo Ren.
So, to conclude: lightsabers do not lose their power when their owner dies. Although each Jedi may have an affinity with a lightsaber they crafted themselves, they may use another lightsaber as a weapon. The Force is the real weapon, lightsaber is just a tool.
Update: The Jedi were rare even in the Republic era. There were less then ten thousand Jedi at most.
As Jedi became a myth (Following the Great Purge aka Order 66 and some years of Empire reign), the lightsabers could be around, lying there in a scrap pile of discarded shipware. Nobody, except a few, would recognize a lightsaber as such and know what is it capable of (in the right hands). A typical scavenger would consider it an antique item of unknown purpose, worth probably nothing.
During the Purge, the Jedi were scattered across the Galaxy, as shown in RotS. Their lightsabers are probably still around, lying near their decaying corpses or under debris. Someone also mentioned the new canon stories where lightsabers of killed Jedi were collected and destroyed for public amusement as part of anti-Jedi propaganda.