Velen's Vision and the Bequest of the Wind
In it he saw the Broken: formerly healthy draenei who had devolved during the orcs' murderous crusade to wipe out the draenei race. According to Velen's vision, the Broken would rise to power once again and aid their displaced brethren.
Velen's vision was given substance in the form of Nobundo, a one-time draenei Vindicator who had devolved while the orcs decimated his race and tore the planet apart. Like his fellow Broken, Nobundo had lost contact with the Light, and so he ventured far into the deserts of Outland to meditate and pray for guidance.
After decades of silence, an unfamiliar voice finally answered his prayers. It was not the Light that whispered to him, but the wind. The breeze spoke to him of lost truths, of the might of the elements – of the delicate balance of power embraced by the shaman.
Nobundo listened eagerly and learned all he could. When he judged the time was right, he departed the desert determined to use this knowledge to help the draenei race. However, most draenei refugees greeted the sight of a Broken with skepticism and outright prejudice. Only one of the healthy draenei did not shun the Broken: the wise and perceptive Velen.
Velen had heard the philosophy of the shaman before, but Nobundo revealed new truths. The elements were timeless; they stretched across the cosmos to inhabit every world they touched. With the proper training, the draenei might learn to wield this overlooked, seemingly inexhaustible source of power.
Velen rejoiced to realize that his vision was coming to pass. Accordingly he took Nobundo with him when he and his fellow refugees took possession of the naaru satellite structure Exodar from the blood elves. Thus it was that Nobundo journeyed with the draenei to the world of Azeroth.
Nobundo found that unlike the desolate wastes of Outland, Azeroth possessed abundant elemental energies. He shared his knowledge of shamanism with the draenei on Azeroth, and all who adopted its path flourished.
The time for the draenei shaman to test themselves draws near, for the battle against the Burning Legion is once again at hand. And the fate not only of Azeroth, but of all worlds, hangs in the balance.
From the Burning Crusade website: 
Shamanism is not popular among the draenei, with many thinking it gets in the way of worship of the light. 
The reasons for dwarf shamans aren't entirely known, but it likely has to do with the alliance with Wildhammer dwarves and Dark Iron Clan to Ironforge. A suggestion is that due to the creation of the Council of Three Hammers the Wildhammer and Dark Iron dwarves are part of Ironforge (faction). As such we will be able to play them as now they are part of Ironforge. This is also true in the case of why Dwarves can become Mages due to the inclusion of the Dark Iron dwarves in Ironforge.
Tauren are perhaps the most spiritual people on Azeroth. They revere their ancestors, the spirits of the land, and the great and vague force they call the Earth Mother. They assisted the Horde in developing their forgotten shamanic roots. Tauren honor their shaman as they do few others, and their shaman in turn keep their communities in touch with nature and reinvigorate their warriors and hunters with spiritual vigor and purpose. The iconic tauren shaman is a calm individual. A meditative spirit holds in check their great size and strength - they are like a boulder in the tide, steadfast and immobile, watching the world as it changes around them. In battle, however they are a terrible force, calling upon the spirits to annihilate their foes.
Orc shamansDuring the Third War
Under the leadership of Thrall, the orcs have rediscovered their ancient Shamanistic traditions. Because of their history, the orc warlocks are barely tolerated and no longer practice their magic as openly as they once did. Now, most orc magic users practice shaman magic which draws its power from the natural world and the elements of the earth. Powerful shamans can call lightning from the sky and rouse the earth itself to devour legions of their enemies.
Shaman can cause such overwhelming bloodlust in their brethren that affected warriors actually increase in size and power for short periods of time. The shamans’ lightning shields, when cast on fellow warriors, create a barrier that damages any nearby ground units. Shaman can purge any magical spells or enchantments from their comrades or enemies with their ability to purge. Though purge can be useful to remove negative enchantments or curses, it taxes the recipient to such a degree that the unit’s movements are slow and painful.
Orcs revere the natural elements and Shamans are held in high regard. They generally have a close relationship with the nature elements and angering them is considered a grave offense. There is probably no finer example of this than what happened after the Stormpike Guard angered the spirits of Alterac Valley by digging through the ancestral grounds of the Frostwolf Clan in order to find alleged ancient relics.
|Source information needed!|
The trolls have always had deep connections to the spirits, though never called it shamanism. The way of the witch doctor, which might after time lead up to the respected title, Shadow Hunter, was a common way of life to many trolls. But not all trolls could comprehend the power. A witch doctor would not only have to be able to strike lightning into their foes, with the power granted by the Loa God Shango, controller of storms, or aid allies by mending their wounds, with the might of the Loa God Lokou, master of healing. They would also have to be clever, cunning and rutheless in order to save their tribe, or stop anyone who came in their tribes way.
When merging with the Horde, they taught their new allies their "twist" on shamanism, and tauren and orcs taught them theirs.
Not much is known currently about goblins as shaman, however it is a confirmed class choice in the following expansion: Cataclysm. They will most likely be taught in the ways of the shaman by the orcs who are assisting the goblins of Kezan in their fight against the alliance.
However as their shaman trainer Max Avalanche is present before meeting the orcs, goblins may instead have learned shamanism from their former jungle troll oppressors who now serve the goblins according to a comment made by Foreman Dampwick and a Cataclysm art piece showing a troll being kept by a Trade Prince.
Goblin shaman are an extension of their society's single-minded devotion to making a profit; to a goblin shaman, elementals are potential customers. Goblins do tend to be a bit more forceful in their negotiations than the other shamanic races (especially the tauren) would like, though they are far less forceful than what we've seen from the taunka in Northrend. (Unless the elemental tries to weasel out of its contract. Elementals tend not to have breakable knees, so goblins sometimes have to resort to other methods of control.) As for the goblins' "mechanical" totems, note that these are merely physical manifestations of the small totems they tinker/craft to form a link with the elemental spirits. Instead of lugging around large totems, goblin shaman have a ring (probably the same ring on which they keep their house and motorbike keys) with small totems they've built as conduits for the elemental spirits they do business with. 
Furbolg society places great importance on shaman. Each furbolg tribe contains at least one shaman, and most of the time a shaman leads the tribe. Lesser furbolg shaman range out with the hunters to protect their territory. Furbolgs share a connection with nature as well; their shaman communicate with the spirits of nature and focus on divine magic that flows in furbolg blood. The iconic furbolg shaman is a peaceful being carrying a feather-topped spear who guides his tribe with benevolence. When his anger is aroused, however, he is a fearsome sight, using his magic to rouse himself and his warriors into a horrible frenzy.
Tuskarr shamans have the power to influence events such as weather, food, and illnesses. Shaman magic is often quite a production, even something like curing the sick: the shaman may speak with ghosts of the patient's relatives and even battle other spirits into submission before forcing them to help heal the patient.
Quilboar shamans possess the abilities to combat or cause disease, to control nature, and to summon spirit boars. According to quilboar shamans, the existence of sickness is caused by the intrusion of other species on the rightful lands of the quilboar. They teach that until the invaders are driven out and the quilboar have returned to their promised lands, suffering can only be alleviated through a heroic death. The afterlife for quilboar martyrs is filled with wonders and pleasure beyond imagining, according to shamanistic teachings. Quilboar shaman teach that the law of the wild is the standard by which all tribal members must live - survival of the fittest. At least one quilboar every generation walks the path of the shaman. Quilboar never seek to become shaman, but a few are awarded the right to study with their elders when visited by dreams of former shaman now deceased. Shaman usually rise to positions of power and leadership within their tribes. Most often it is males that receive the dream-vision that sets them on the shaman's path, although female shaman are not uncommon. On occasion, a quilboar receives a dream-vision where the deceased shaman of the tribe counsel him to leave the pack. The vision may seem confusing to the quilboar, but tribal shaman take these visitations seriously.
Dark Iron dwarven shaman
The spellcasters of the Dark Iron dwarves are usually mages or shaman, as these deal closely with fire.
Wildhammer dwarven shaman
Shaman and priests of the Wildhammer dwarves entertain people on cold nights with myths of nature, and the Earth Mother. The Wildhammer dwarves lead much more spiritual lives than the Ironforge dwarves. They are dedicated to elemental earth and storm magic, drawing their power from the mountains. Magically, they are more powerful than their cousins; they have mastered the art of divine elemental magic and use this power to supplement their already mighty aerial fighting force.