The image of the orc warrior calls to mind the blood-red-eyed growling monstrosity roaring with demonic bloodthirstiness and wielding his great axes with abandon in his single-minded pursuit of nothing but death and destruction. For a time, while under the control of the Burning Legion, many orcish warriors were like this, but now even the wildest of barbarians remembers the need to rein in the anger from time to time. Orc warriors (or, at least many of them) are a civilized people when all is said and done, though you might not know it if you only saw them in battle.
Grom Hellscream is probably the most famous orcish warrior, a blademaster who combined immense skill with unbridled passion for the fight. Durotan, Thrall's father, was the very image of the restrained warrior who put the reasons for fighting above the passion of battle itself. High Overlord Varok Saurfang is the perfect role model of the aged warrior who regrets the excesses of the past, and makes up for it with special wisdom and caution in the present.
Trolls are most likely to blend voodoo beliefs and ritual in with their battle. Their religion is savage and primal, and it fuels their fury with beliefs about revenge, death, and the desires of the spirits. They do not have the demon-possessed history of the orcs, but they are sometimes equally vicious. They are well known as berserkers who get more and more dangerous the more wounded and cornered they become.
The most famous troll warriors are Vol'jin, leader of the Darkspear tribe, and Zul'jin, ruler of the Amani trolls in Zul'Aman, although both of them are witch-doctors in addition to being warriors. Vol'jin is by far the more approachable of the two, known for his wisdom and intelligence. Zul'jin on the other hand is well known for his all-consuming hatred of the blood elves and anyone associated with them. Any troll warrior is likely to follow their example of intense cunning, likely with a good mix of voodoo rituals and ruthlessness thrown in too.
Despite their immense size, and the apparent limitlessness of their battle fury, the tauren are among the least likely to be totally consumed by rage in battle. At all times they are connected to the earth and are intensely aware of practical concerns as well as bloody ones. A tauren warrior's sense of spirituality and religion is almost diametrically opposed to that of the trolls, and he fights to bring natural order to the chaos, even if outwardly he seems like a giant savage monster to some.
One of the most famous tauren warriors is Tagar, Cairne Bloodhoof's second in command, who helped guide and command the tauren in times of crisis. Cairne himself could also be considered a warrior, though he may also be a druid or something else instead.
Many Forsaken warriors remember a lot of what being a human or elven warrior used to be, with all their talk about honor, justice, the glory of Lordaeron and all that. They may have been called Knights or Champions at one point, or they might have just been regular old guards and soldiers, but now all that has faded away for most of them. Among the Forsaken, great warriors may be known as Deathguards [Thanks Thiosion!].
Who's to say what really remains? Do Forsaken warriors feel anger and rage the same way a living person might? Undoubtedly, many are driven by a desire for vengeance against the Scourge and perhaps even humanity as well. But this rage likely does not burn in them the same way that an orc's might -- rather it is cold and calculating. An undead warrior's rage not unbridled passion or anger so much as it is almost intellectual, so very sharp and chilling that it cuts like a scalpel.
Of course there are blood elf warriors in the Warcraft setting, but for various reasons none of them are player characters, so we won't touch on them in this article except to suggest that someone who wants to play a warrior-like blood elf choose a paladin, a rogue, or perhaps best of all, a death knight, for a similar type of character archetype.
Also, keep in mind that none of what I say in this or any other article is the one-and-only truth. This is not the one true way to roleplay -- rather it is a proposed standard based on lore and common sense, to which you are welcome to contribute your own ideas and or disagreements in the comments below. These are places to begin when you think about how your race and class interact to make your character who he or she is, not limitations I'm trying to impose upon your creativity. Perhaps these articles can help you think of a reason why your character is different from the standard case. In fact, many of the most interesting characters will break these molds even as they to conform to the same lore and common sense they are based upon.