Update: 29th May 2014
The late essay Of Dwarves and Men contains what seem to have been Tolkien's final thoughts on the matter. Christopher Tolkien dates this to at least 1969, based on strong evidence ("It was written on printed papers supplied by Allen and Unwin, of which the latest date is September 1969") and the bulk of it was printed in History of Middle-earth 12, but the relevant section was extracted to form part of the Drúedain material in Unfinished Tales.
An author's note (note 5) to this material states:
To the unfriendly who, not knowing them well, declared that Morgoth must have bred the Orcs from such a stock the Eldar answered: "Doubtless Morgoth, since he can make no living thing, bred Orcs from various kinds of Men, but the Drúedain must have escaped his Shadow; for their laughter and the laughter of Orcs are as different as is the light of Aman from the darkness of Angband." But some thought, nonetheless, that there had been a remote kinship, which accounted for their special enmity. Orcs and Drûgs each regarded the other as renegades.
This, of course, never made it into the main stories and was ommitted from the published Silmarillion because of that.
The only final interpretation that seems reasonable to me is that:
- There is no known "true origin" for Orcs within Middle-earth; what we're presented with is a bunch of in-universe theories.
- One of those theories is that Orcs were bred from corrupted Elves.
- Another is that they were bred from Men.
- And some believe that the Drúedain were a likely candidate.
Original answer - February 2013
There are a number of problems with all of the origin stories that may explain why Tolkien never settled on a definitive one himself.
If Orcs come from corrupted Elves, are they immortal within Arda? What happens when they are killed? Do they go to the Halls of Mandos, and can an Orc repent and be restored to his Elf-dom?
If Orcs come from Men, what of the timeline problems? How did Morgoth manage to breed so many in such a short time? You can see Tolkien grasping in this direction with some of his later revisions, moving the origin of Men further back, for example.
Problems with Morgoth not being able to create life himself have already been discussed, but at a stretch could be handled by the story of Morgoth disseminating his power throughout Arda.
Ultimately this was never resolved so there is no "true" origin. Orcs must be viewed in the same light as Tom Bombadil; one of the world's unknown and unknowable factors, and we must be content to leave it so.