I wonder if the cause of Jeong  Woo's memory loss could be a hypnosis?
Maybe someone hypnotized him into forgetting or committing the crime and so if he forgets he wouldn't/couldn't) even try finding out/revealing the real killer.
But to forget everything regularly it must be something psychological right? Something his subconciousness wants? Maybe bc he witnessed the death of his family and it gave him a shock or maybe he really did do that? Like on drugs? Hmm... 
Its confusing bc we don't really get the hint that Cha Min Ho who should be our main perpetraitor  has something to do with his memory. He was surprised when hearing that Jeong Woo doesn't remember. 
but cha min ho is always watching him and hope his memory didn't return. or maybe it was just because of his old case? 
I've wondered the same thing, but I've also had the thought that possibly he is forgetting because his friend is somehow involved. If the memory is that painful every time he gets to the point of remembering he could block it all again. I may be completely wrong but I think jos friend is involved in this a lot more than just turning against him because he believed he murdered them. He has already admitted to falsifying evidence.. this is one sketchy dude in my opinion.  It could also be a hypnosis thing where a specific sound or action triggers it.. I've never really heard of that triggering memory loss but I have heard of it being used for other things like in the drama where a certain thing was used to trigger a shooting.. anyway, I know I'm far reaching here but I believe 100% in his innoncnede so I'm ruling out nothing!! So glad he has this lawyer now, she's going to be his redeeming factor. 
I suspect from the psychotherapist the criminal psychiatrist from the first moment. 

What might happened is he had a trauma and he forget his memories, I believe he has a twin or someone pretending to be him wearing a mask of him, we even saw this happening in historical dramas, we had it Mission Impossible way back when it was a TV Drama in early 1990 late 1980's. 

Hypnosis is quite possible, drugging is also quite possible.  For drugging if it is drugging someone in his close circle is doing it, and it also needs to continue in his prison cell..   

For the framing I suspect from the chief he is related that's what I am guessing not everyone is as honest as him
Now we know that the murder happened between 12 something pm and 1:30am, within 30 munites and EVERYTHING points out towards Jeong Woo's freied since he was there. but it just got my head spinning. what possibly could've happen in that 30 minutes that killed Jeong Woo's wife?! I mean it just doesn't seem that someone could've broken in. And I also kinds don't believe his friend did this to her too. He loved her.
A lot of good theories here. I just watch episode 6 and I wonder why no one in the drama questioned the context how the confession was obtained. i.e. was he answering or questioning the Prosecutor's accusation? Was he drugged at the time. Why not analyze the entire footage before putting someone on death row? Also, shouldn't there by a written confession or sign off after the verbal? Just my 2 cents here. Sorry, I watched too much American crime TV :)