Bonaventure, as I understand, it is not ALL organs, but only single unpaired organs, which must come from a live person.
Though I would think that the fresher any twinned organ, say a kidney, the better would be the result. Which would be a further enticement to kill for a greater success rate.
Not to derail the thread, but.... only live organs can be harvested, regardless if they are paired or not. Other non-organs can be harvested after death (e.g., bones, cartilage, ligaments), but not organs.
Here's a little blurb from
this article:

A couple of years ago I attended a talk on this subject by some old and wise doctor who, based upon Catholic principles, was adamantly opposed to organ donation (regardless of the Church's present stance on it). Essentially, he pointed to the high rate of organ refusal by the recipient, concluding that for the most part, God did not intend us to be swapping organs with one another. It was also at this time that I learned that, despite what I was told when I signed up to be a donor when getting my license, organ donors had to be alive.... at least from a blood circulatory sense... in order to harvest organs. Once blood circulation stops, doctors have between 3 and 5 minutes to harvest the organ, otherwise it is "dead" tissue and can not be used in any transplant. So, if you have "donor" on your license, and you're in the hospital and things aren't looking too good... maybe you're in a coma, maybe you're "brain dead"* but still showing some activity, how much trust are you going to put into your doctor who may be looking at your with $ signs in his eyes? Don't you think there may be an incentive to "pull the plug" prematurely if they can harvest your organs immediately prior to doing so?
So.... going back to the example of the 'murdered' donating organs (not including murdered babies in the womb), but none of them are ever organ donors. Not unless they had a doctor standing over them when they were "murdered"! (Probably happens in China a lot). Same thing with car accident victims, etc.
* I also recall this doctor saying that there was no such thing as being "brain dead" and that that was a term that had come into the lexicon primarily to further organ harvesting.