As another answer shows, after Anakin defeats Count Dooku in a lightsaber duel at the beginning of Episode III, he drops Dooku's lightsaber and leaves it there. He doesn't just ignore it, he specifically drops it, as he used it to kill Dooku. Why does he do that? It would have been a proof of what he's done, in case someone (the Council?) asks for it. It's also a dangerous weapon (that shouldn't be left around), it's easy to carry, and it's even valuable1.
On the other hand, at the end of the movie, when Obi-Wan wins the duel, we see that he picks up Anakin's lightsaber.
What do Jedi Knights do with the lightsabers of the enemies they have defeated? Why doesn't Anakin take Dooku's, whereas Obi-Wan takes Anakin's?
1 According to this answer on Reddit,
Canon: In Season 2, Episode 11 of the Clone Wars, the black market price of a lightsaber was given at 20,000 credits. Season 5 Episode 6 tells us that the main components of a lightsaber are easy to come by, but it is the kyber crystal that makes sabers so costly and rare.
To put it in perspective, in Episode IV Han Solo demands 10,000 credits to take Luke and Ben to Alderaan, and an outraged Luke says that "We could almost buy our own ship for that!", which indicates that it's a lot of money.