The cleric is sort of an odd duck in D&D. The fighter, the thief, and the magic-user classes can all be described in terms of their favored method of problem-solving in the game. The fighter uses force, the thief wit and stealth, and the magic-user magic. One could say that the cleric solves problems with "faith" or "divine inspiration" or some such thing, but that's just obfuscating the fact that his methods are a mixture of fighting and spell-casting and power over undead monsters, and thus isn't really a niche unto itself.
The defining quality of the cleric is a cultural role, not one of methods. Specifically, the cleric seems to be a quasi-historical model of the Christian crusader, and things sometimes get a little awkward when trying to use the class to represent very dissimilar faiths. AD&D 2E tries to resolve this problem by dividing the spell list into various "spheres" and allowing specialty priests, who may have different armor and weapon restrictions, other powers in place of turning undead, and can access various of the spheres of spells. That's one way to do it, but it's pretty fiddly for classic D&D. Classic D&D did give this a shot in the Wrath of the Immortals boxed set, with a long list of the various immortals and the weapon and armor restrictions and special perks of their clerics, though they all used the same spell lists.
I don't really want a hundred different sub-classes of clerics in my game, but a little bit of distinction would be nice. So here's how I'm going to break it down, based on broad categories of religious beliefs rather than deity-by-deity.
Human-centric: These faiths may be monotheistic or have a pantheon of gods; their salient characteristic is that the god(s) treat humans (and/or other races) as their children, to be shepherded and guided toward fulfilling their part in some grand and inscrutable divine plan. The god or gods may be, or purported to be, the creators of humanity or of all existence. Humans are afforded a place in the cosmic order above mere nature, as stewards of the earth. The D&D cleric as written represents this type.
An intolerant One True Way religion is likely to be of this type, though tolerant versions are certainly possible as well. A human-centric faith is probably going to be either good or evil; it's hard for me to conceive of a neutral example. An evil religion of this type treats humans as the slaves and pawns of the god(s), livestock rather than children, with the priesthood acting as taskmasters administering the god's will, attaining special status among the god's thralls by submitting themselves willingly to its service. The good and evil religions may be counterparts, with the evil god or gods having fallen from grace. They may send their followers to infiltrate and undermine the followers of good, or tempt good clerics into sinful behavior, and thus may the Great and Holy Church become corrupted. A priest may still preach the word of good with a false tongue, while continuing to receive spells from his true master - perhaps knowingly, or perhaps completely unwitting and believing that he still serves the cause of good!
Nature-centric: Religions of this type may worship a pantheon of gods representing the natural world, or a single god or goddess representing nature as a whole. Man is generally held to be a part of nature, not apart from it nor above it. Clerics of these faiths may see themselves as representing man's interests in the natural world, performing rituals to protect their communities from the vicissitudes of nature, ward off plagues, ensure bountiful harvests, etc, or they may represent nature itself and seek to prevent any side or force from upsetting its balance.
Traditional D&D druids fit this role, as do shamans of many primitive societies. Nature-based faiths are particularly prone to be neutral, but good and evil-aligned ones are possible. A good one upholds man's rightful place within nature, and seeks to help him live in harmony with it. An evil one might see man as a scourge to be subjugated or eradicated.
Nature clerics use the druid spell lists. They are limited to leather armor and leather or wooden shields. Weapons should be predominantly of natural materials such as wood, bone, or stone - spears, clubs, bows, and staves are the most obvious examples. Metal may be strictly prohibited, or simply eschewed in favor of natural materials whenever feasible. Turning undead really isn't the province of nature priests, but turning of normal animals would be appropriate, with "turn" results handled normally and "destroy" results taming the creatures instead. (See here for a simple turning mechanic that uses Hit Dice rather than a chart of specific creatures, and so is easily adaptable to any creature type.)
Indifferent gods: Typically a pantheon-based religion. The gods and goddesses are generally ambivalent toward humanity, having their own melodramatic affairs to occupy their attention, but often find mortals useful or intriguing enough to pay some attention to them. They may seek worship for reasons of vanity, but their favor is fickle.
The ancient Greek and Roman mythologies are examples of this type of religion. They tend to hew close to neutrality overall, but due to the capricious nature of the deities they may often skew toward good or evil.
In general, clerics of such a religion are unlimited in terms of arms and armor. They use the standard spell lists, but each day when they memorize spells (or their spell slots or points are restored) a reaction must be rolled to see if they have the gods' favor. On a neutral or better result the cleric gets his spells as normal. On a hostile result the gods grant spells at only half the cleric's level. Once a cleric falls into disfavor, he or she must perform a sacrifice or some other ritual to appease the god(s) offended. Optionally, the ritual may be performed pre-emptively, granting a bonus to the reaction roll, to ensure the favor of the gods before undertaking an adventure. Turning undead seems appropriate to clerics of this sort; spirits escaped from the Underworld must be driven back whence they belong. If the gods have immortal rivals to their power, then perhaps clerics can turn the minions of the rivals instead (though they may well include the undead.)
If desired, some individual gods and goddesses within the pantheon may cultivate closer relationships with their mortal followers and function as humanistic or nature deities instead.
Philosophical, non-theistic*: No gods are worshiped; rather, clerics of this sort of religion seek balance, enlightenment, and oneness with the universe from within themselves. What exactly that means and how it is pursued may vary from one sect to another, but meditation, introspection, austerity, and strict self-discipline are common methods in lieu of praying to the gods.
Buddhism is the most prominent real-world example that comes to mind. These religions tend toward harmony and benevolence toward self and others, which would qualify as good by most standards. However, it's possible that one might exist which teaches that enlightenment comes through transcending of restrictive moral standards that bind "lesser" men, or through experiencing and inflicting suffering, or some other less than savory means.
These clerics do not wear armor or use shields, but are unrestricted in their choice of weapons. (Most will likely choose something in harmony with their philosophy; quarterstaves are a popular choice among those whose outlook is peaceful and benevolent, for example.) A new and novel spell list is probably in order, though I haven't prepared one yet, but spells allowing feats of mind over matter would be favored. Turn undead seems inappropriate, but perhaps the ability to "turn" illusions and mind-affecting magic fits better. Treat the illusion as a creature with HD equal to its caster; if caster level is unknown, use the lowest level at which the spell can be cast. Turning allows the illusion to be disbelieved or the spell to be resisted; destroying allows the cleric to assert control over an illusion or turn the spell back on its caster.
*AD&D fills this niche with the monk class, and the Rules Cyclopedia adds the mystic class to basic D&D. I prefer to keep things more within the purview of the cleric class rather than adding a new class of martial artists. Your mileage may vary, though.
A few more stray thoughts:
Within each of these broad archetypes will be found sects espousing differing beliefs, and if desired they can be distinguished from one another through non-mechanical details, such as specific codes of dress and conduct. Weapon restrictions can be freely changed up to achieve the right feel for a particular sect. The standard cleric is limited to blunt weapons, but allowing one order to use spears and javelins instead of maces and hammers doesn't change the balance of power among the classes at all - those clerics are still limited to 1d6-damage weapons. Clerics belonging to orders known for specific talents, such as healers or oracles, may distinguish themselves through the spells they choose rather than by having special spell lists, in much the same way a fighter can be a swashbuckler or a barbarian by his choice of arms and combat tactics rather than special rules.
One other possibility that occurs to me as a source of clerical power is some sort of eldritch abomination or Outer Being, as are abundant in the Lovecraft mythos, for instance. Priests of such cults probably ought to be treated more as monsters than player-character classes. Their powers need not be constrained by the usual concerns of class balance, and should be alien and disturbing to players.