As you may have guessed, the plot to Amnesia is that you have amnesia—that little story-telling chestnut. If adventure gaming were a medical condition, the first symptom would be amnesia and the second would be kleptomania.
Wherein the hero has amnesia. He can't remember a thing, except oddly his own name. Sometimes, although rarely, the hero cannot remember his name. Then someone has to say "Well, we have to call you something!" and they have to make up a name for themselves. (Unless the hero is established to us beforehand - in which case he will often unsuspectingly choose his old name, perhaps to avoid confusing the viewers.) Usually, the Amnesiac Hero:
Has amazing fighting skills, but no idea how he got them. This makes his origin even more mysterious.
Is found by a handsome/beautiful soon-to-be sidekick, who helps them on their journey to remembering who they are.
Eden of the East if you don't count the girl as the hero. The person in question shows up with no clothing and no memories, nothing but a cellphone and a gun.
Alice from Pandora Hearts fits this to a "T". She's the most powerful chain in the Abyss who knows only her name, and she inevitably goes on a quest to find her lost memories (though she soon discovers that she may have forgotten them for a good reason. Then she discovers that the painful memories she found were actually those of her twin sister's, and that she once tried to destroy them. Too bad her attempts caused her to forget, too).
Oz is this, too. He seems to have forgotten his time as B-Rabbit until Jack showed him the truth.
Mikage from the manga Torikago Gakyuu is this as well, although he doesn't even remember his name. He's given only the surname "Mikage" by a teacher who promptly reveals to him that he is from a parallel world.
This was Wolverine's schtick for many years; that he couldn't remember his Mysterious Past, yet kept running into people that were from it. House of M cleared this and had him remember everything, which lead to the discovery of his son Daken.
In "The Groundstar Conspiracy" a desperately wounded man staggers away from a sabotaged lab in which several other people are found, so incinerated that they will never be identified. Is he the saboteur or an innocent victim? When they repair his wounds and reconstruct his face (based on the ID badge he was wearing, which may not have been his) - he claims to have amnesia and escapes to avoid being locked away for life. Eventually we find he is not the saboteur, who died on the operating table, but bait put out there to capture the rest of the spies. His permanent amnesia was induced.
In RoboCop, Murphy forgets everything about his past life after his death and resurrection as the titular cyborg crime-fighting superhero.
In Dark City, John Murdoch wakes up in a hotel bathtub, not knowing how he got there, what his name is, or why there is a mutilated hooker on the floor. He only finds out his own name when he tracks down his wallet at a nearby automat. The reason that John has no memory is because his and everyone's memories have been manufactured by the cabal that runs the city.
Rachel is this in the first Megamorphs book of Animorphs.
Gentle in Clive Barker's Imajica, albeit intentionally self-inflicted. Lampshaded by a character who knew him pre-amnesia: "I was told you'd changed, but not how much."
Andrei Belyanin's Jack the Mad King trilogy has this as the plot of the first novel. In fact, while Jack eventually discovers his identity as the true king of the land, he doesn't recover his memories and chooses to go on adventures instead of deposing his brother, who has taken the throne after believing that Jack is dead (the brother's wife was the one who tried to kill Jack). The lack of memories is barely mentioned in the subsequent novels, as Jack has accepted his new identity.
This trope, along with his knowing everything, was the premise for the short-lived series John Doe.
Slightly subverted in that it wasn't the character's pre-amnesia life that granted him ultimate knowledge but the process itself. This is explained by Word Of God after it was clear that the show would not be renewed. Actually, one episodes does attempt this explanation by a villain, only for it to be revealed as a lie.
Doctor Who - The Eighth Doctor's delayed regeneration leads to an especially bad case of Resurrection Sickness - Waking Up At The Morgue with no idea who he is. He gets over it, only for this trope to follow him like the plague in his many expanded universe adventures. Whether it's the audios, magazines or books, he loses chunks of his memory with worrying frequency.
Slayn Wilder, the protagonist of Growlanser: Dual Darkness begins the game with amnesia.
In Neko Kawaigari, the player is a dog-man named Jack who, you guessed it, has amnesia. He's rescued by the head of a clinic for catgirls. In order to repay her, he becomes an odd-job man at the clinic, helping with the patients.
Pokemon Mystery Dungeon: Excepting the Japan-only WiiWare games, the player is a Pokemon that was once human and lost their memory and in the first game it was more or less self-inflicted. However, unlike a normal Amnesiac Hero who is an inherently good fighter, they do have to slowly learn fighting skills (Pokemon moves) the same way any other Pokemon does: by Dungeon Crawling and Level Grinding.
The Nameless One from Planescape: Torment, except he (obviously) doesn't remember his own name (his name is the last thing he remembers). Oh, and the soon-to-be sidekick who finds him is a floating talking skull who didn't so much 'find' him as was waiting for him to wake up again. However, whether he qualifies as a hero or not in this game depends on the player.
Daniel from Amnesia The Dark Descent with a twist; he induced it himself so he wouldn't remember the horrible things he did.
The Warrior of Light in Dissidia: Final Fantasy is an interesting case. He tells the heroes he has no memories of his past, not even his name, and hopes to someday remember. However, it's revealed this is because he has almost no memories to remember — unlike the other warriors he wasn't brought into the war from another world, he was created during their proceedings, and the memories they've lost at the end of each cycle, for him are the only memories he's ever had. That he's able to remember anything at all is proof of his Character Development from a blank slate with no will over the course of the cycles.
Sissel in Ghost Trick. Of course, all ghosts in the game suffer from amnesia until they remember who they are; it just takes Sissel much longer than everyone else. It doesn't help that he's trying to recall the memories of the wrong person the entire game.
The hero of Arx Fatalis has no memory and is quickly given the name "Am Shaegar", which means "The Nameless One". He actually gets his memory back relatively early in the plot; the amnesia really seems to be there just to justify the hero having to learn the nature of the world.
Fei from Xenogears. He wakes up in a village with no memory of what he was before that, and all that he can conjure up from his past life is his martial art skills.
Luke von Fabre, main character of Tales Of The Abyss, had his whole memory wiped when he was kidnapped at the age of 10 and had to relearn things like walking, eating and not wetting himself. He is actually a clone of the original and was created during that kidnapping. As such, his memory wasn't so much "wiped" as "non-existent" when he was sent home in place of his original.
Super Robot Wars UX: According to Fei-Yen HD, she was in a dark area all alone until she heard a voice. Upon reaching it, she became "Miku". What triggered the transformation was the whole concert with Sheryl Nome, which made her excited when she heard the word "concert" & "song", but what reactivates her memory is an attack by the Vajra, which gave her the "shock treatment" necessary to bring her connection (while vague) with Miku. As a result, Grace O' Connor is targeting her as a potential nominee for her plans.