The Italian fairy-tale author Giambattista Basile wrote a number of stories which featured a creature he called "huorco" or "huerco" - probably derivatives in the Neapolitan dialect of the Latin underworld deity "Orcus." This is one of several probable etymological sources for J.R.R. Tolkien's orc, filtered through Ludovico Ariosto's Orlando Furioso, where the form became "orco." It went into French through Charles Perrault, who used the form "ogre" for the same creatures. In so doing, the ogre became the popular form of the large, hairy beast who terrorized the heroes of fairy stories. Orcs went down a different road, becoming in Tolkien twisted creatures derived from elves. As is often the case, when it came to D&D, Gygax split the difference and added both to the roll of monsters. Orcs became fodder for a million fighters' swords, and ogres were considerably larger and stronger. At least in this post I want to stick with the orcs.
It is hard to deny a certain old school appeal to David C. Sutherland III's orcs. When you describe pig-men, you know these are orcs and their job is to try to stab PCs, and get stabbed in return. There's no horror to them, no moral equivalences, it's a tried and true foe that does perfectly well to say "we are in D&D fantasy."
OD&D simply describes them as tribal creatures which hate daylight like goblins do, spending more time worrying about the kind of fortifications of their lair than what the things actually look like - one of the more refreshing aspects of the OD&D books when you are interested in re-interpreting monsters. They are merely expected to live in tribes that act as the foils of human civilization; their particular organization usually revolves around some more powerful opponent. An individual orc is exactly at the point to go toe-to-toe with an average PC of first level; all this tends to make them humanity's dark mirror. If you go back to the actual play report from a convention game run by Dave Arneson that I had put up several years back, orcs are assumed to be encountered in masses in the early dungeons.
My inclination is to run with the bestial undercurrent of the orc. If you think about "beast-men" in roleplaying, several are obvious. The classic kobold is a dog-man, the gnoll is a hyena-man, the minotaur is a cow-man, the rakshasa is a tiger-man. Sutherland's orcs are pig-men, and I think that's fine - if it's one tribe of orcs. This feeds into something I've been pondering for a while, about a question of diverse and interesting creatures that resemble humans with various animal heads.
In keeping with the loose inspiration of Orcus, a human with a goat head seems to give the almost Satanic presence that is demanded by the name of an orc. Just the slit-shaped eyes should seem freakish. A donkey-man brings home the idea that this is a beastial creature (and puts us in mind of fairy tales and A Midsummer Night's Dream), and perhaps allows a bit more variation; is this a more "neutral" tribe of orcs, but one that takes on the stubbornness and ill temper of the creatures they resemble.
This is a question of "freshening up" monsters that I think is important to find a certain balance. With the creature list from Monsters & Treasure, which I think provides the majority of the "true" classic D&D monsters, I think there's a need to balance between what is familiar (sometimes to the point of cliché) and novelty merely for the sake of novelty. A deeply tribal nemesis like the orc is a perfect thing to play with: your "normal" pig-faced orcs are one tribe among many, and there are weirder orcs. Particularly this allows the freedom to really play with different tropes. The pig-faced tribe is your standard "raid and retreat" sort of orc, while the goat-headed orcs are devout Satan worshippers and perform vile rituals, and the donkey-headed tribe are open to negotiation but no friends of humanity.
Other types of orcs open for interpretation might base themselves on more or less sinister or vicious animals. I think there's something to really distinguish some of your orcs while keeping the good old familiar ones around.