Clerics are a rare and unique phenomenon among the demi-humans. They do exist, but they are so rare and specialized, they are not available as player character classes.
In a very interesting table in the DMG, on p.12, it is revealed that these demi-human clerics do not receive their calling until they are fully mature, at the beginning of the "middle age" category (gnome clerics being a minimum age of 303, dwarfs 252, and elfs 510, no age given for the halflings). The PH, pg. 14, reveals that they have level limitations as well (gnomes 7th, dwarfs 8th, elves 7th, halflings 6th -as druids only though).
Essentially, demi-human clerics are like tribal shamans or witchdoctors. As revealed on p.14 of the DMG, shamans/tribal clerics of even the advanced humanoid races (giants, goblin, hobgoblin, lizardmen) are limited to the 7th level. The lesser humanoid races are limited to the 5th (bugbear, gnoll, kobald, orc) or the 3rd (ettin, ogre, trogs, trolls).
[The important aspect of reaching 7th level cleric, is that it gives you access to a 4th level spell. Being a 5th level caster gives you access to 3rd level spells, and being a 3rd level caster gives access to 2nd level spells.]
The level limitation on halflings seems arbitrary, being 6th rather than 7th like the other races, but keep in mind, being a 6th level druid provides access to 4th level spells, so the intended effect is the same. The overall point being, the demi-human and humanoid races can go no higher than 4th level spells.
Primitive Spell List
Shaman spell lists are a bit shorter than the full cleric lists, limited to the "primitive" powers:
1st --> cure light wounds, detect evil, detect magic, light, protection from evil, resist fear
2nd --> augury, chant, detect charm, resist fire, snake charm, speak with animals
3rd --> cure blindness, cure disease, dispel magic, locate object, prayer, remove curse
4th --> divination, exorcise, neutralize poison, tongues
Witchdoctors are considered a bit different than shamans, since they combine clerical and magic user powers. Witchdoctors combine the cleric spells above with the following magical powers as well:
1st --> affect normal fires, dancing lights, identify, push, shield, ventriloquism
2nd --> audible glamer, detect invisibility, invisibility, levitate, magic mouth, scare
The maximum level of the tribal witchdoctor is again determined by race, with primitive humans (cavemen) and greater humanoid races limited to the 4th (goblin, hobgoblin, orc), and the lesser races limited to the 2nd (bugbear, gnoll, kobald, lizardmen). Meaning, the greater races have access to the 2nd level spells, while the lesser races can use only the 1st level spells.
"Look and Feel" of Primitive Holy Magic
Cure light wounds, cast by the various races, would NOT look like a Paladin, eyes closed, murmuring a soft prayer, laying his glowing hands on someone. Each race should have a unique "look and feel" to their spells, that is appropriate to their race. For example, considering cure light wounds:
--Cast by a gnome, it might be a powder blown over a person's body...
--Cast by an elf, it might be an herbal salve spread over the wounded body part...
--Cast by a dwarf, it might the banging together of two rocks in a circle inches away from the person's body...
--Cast by a halfling it might be a root placed in the person's mouth...
--Cast by a goblin it might be symbols drawn in chalk over the person's body...
Cure light wounds, cure blindness, cure disease, remove curse, exorcise, neutralize poison... they might all look exactly the same! In other words, the healing ritual might look exactly the same, the only real difference being the power of the healer, or the amount of time it takes to complete the healing ritual.
A primitive healer wouldn't think to himself, "gee, I'm only 3rd level, I don't have Cure Disease yet..." He would try his healing mojo out, do the best he could, and it would only be revealed in time if the disease was cured or not.
Creation of Sacred Power Objects
Tribal holy men would be experts in primitive magical relations and the charging of objects with natural mana. Notice, this is magic, and it can approximate miracle, but it is not reliant on some extra-planar pantheon of gods. Demi-humans and humanoids don't have astral bodies, and they don't have "disembodied" gods.
They are creatures of magic, embued with the magical mana of life, but of a natural life only. All of their powers come from their channeling of this naturally-existing mana power. It is this magically mana that their holy men are adept at tapping into, especially through the creation of power objects.
Elf holy men, for example, might be in charge of a sacred font of liquid, which is supercharged with mana from a precise mixture of sunlight, moonlight, starshine, leaves, fruits, herbs, magic ingredients, and prayer.
Kobalds, on the other hand, might create their magic stones by choosing a special kind of gem, then charging it up with mana by placing it under a small underground waterfall (which they would call sacred, or magic).
Lizardmen might create create their sacred staffs by soaking them in the blood of sacrificial victims, then drying them in the heat of a dragon's nest. Pulling the staffs out of the dragon's nest, viola! they have mana-charged staffs which allow them to produce various clerical and magical spell effects.
The more intelligent and advanced races have access to higher-level spells, because they are more capable of understanding and applying the "rules of magic".
The hard part for us, as modern players, is to wrap our minds around "religion" that isn't supernatural. Yet that is exactly what primitive religion was: naturalistic. It was a form of power embedded in the material world.
In game terms, this power is of two sources: the positive energy from the sun, called mana, or the negative energy from the astral plane. Matter and soul, dualism fused in a unity.
--Some creatures, like the animals, demi-humans and humanoids, have a material nature, lacking an astral component.
--Some creatures, like humans and elementals, have a dual nature, astral spirit animating a material form.
--Some creatures, like devas, demons, and devils, have an astral nature, although they can be "embodied" under special circumstances. The presence of an astral body is usually indicated by psionic powers.
In other words, the "clerical" powers of the demi-humans and humanoids are not "granted by prayer". They are an expression of their own magical natures, and the ability of their gifted individuals to accumulate and focus the mana energy that surrounds them.