In Diablo I, there were 25 different shrines (not including replenishment). They did odd, dangerous, often permanent, and sometimes negative things. I will not delve too deeply into their functions though, as they are too different from the shrine system in Diablo II and III to make much of a point, but I will make a point that there are 25 unique shrines, some with multiple functions. You can see a list of these shrines and their functions here: http://www.diablowiki.com/Shrines_(Diablo_I)
In Diablo II, there were 16 different shrines (not including replenishment). The majority of them were buff-giving and almost identical to how they are in Diablo III. But let's be honest here. Even in Diablo II, shrines were not that fun. The only two shrines that anyone got excited about was the Experience Shrine when it was inside the Chaos Sanctuary or Worldstone Keep Level 3 and the gem shrine (thanks to those who reminded me of its usefulness). Maybe even the Skill Shrine if it was in the Blood Moor and people wanted to BM pvp. Other than that, the shrines didn't really have a point. You can see a list of these shrines and their functions here: http://www.diablowiki.com/Shrines_(Diablo_II)
Aside from Health Pools, which is the only replenishment shrine in the game, there are only 4 shrines. I feel like the 4 shrines that do exist are just streamlined and consolidated of the buff-giving shrines from Diablo II. Let's go a bit in depth with them.
The Protection Shrine is like the armor, poison res, lightning res, cold res, and fire res shrines all wrapped into one. It's a good shrine for most classes and specs, as most will take some sort of damage at some point.
Probably the best shrine in the game, it is really a more streamlined version of the Combat Shrine, except for giving all classes a damage benefit. However, some specs are actually harmed by attack speed.
From 1-60, this is probably your favorite shrine. However, it is removed from Inferno mode, leaving only 3 shrines for the majority of people's gameplay experience.
This is probably your favorite shrine at 60+. However, once again, it is being made obselete by a magic find cap which over time leaving the game with a total of 2, count them 2, shrines.
The way that shrines are now is just poor design. They are not fun nor are they exciting. When I see a shrine I want to be excited about what it could be and then have fun with its function. The way that it is, I take a shrine when I see it just because there is really no reason not to, but it serves for no real addition to the game. Even though it is a small mechanic, it is nevertheless no excuse to make it a boring one.
The solution is quite simple; add many, many more shrines that have fun functions. Revolutionary, ain't it? I can even get you started! Here are some ideas of shrines that could be added:
Regeneration Shrine: Your character regenerates x amount of resource per second.
Quick Shrine: Your character gains 15% movement speed.
Deadly Shrine: Your character does 25% more damage.
Summoning Shrine (Similar to Diablo II's Monster Shrine): Summons a champion or elite pack at your location.
Healthy Shrine: Your character regenerates x life per second.
Toxic Shrine: Your character emites a cloud of poison around themselves.
Chilling Shrine: Your character emits a frost nova every x seconds.
Fiery Shrine: Your character leaves fire in their wake.
Shocking Shrine: Your character shocks all nearby enemies every second.
Static Shrine: Your character gains +50 gold pickup radius.
Gem Shrine: The shrine drops a gem ranging from chipped (common) to radiant star (extremely, uberly rare).
Powerful Shrine: Your character becomes temporarily invulnerable.
Wealthy Shrine: Drops mass gold.
Lucky Shrine: Drops x amount of items, base rarity being magic.
(These previous two would replace the fortune shrine, which should be removed.)
Elusive Shrine: Increases your dodge chance to 75%
Enigmatic Shrine: Increases a random stat by 200% and reduces the rest by 75%.
Durable Shrine: Repairs all items for free.
Greedy Shrine: Summons a Treasure Goblin at your location.
Colorful Shrine: Drops 6 dyes of the same color.
Knowledgable Shrine: Drops x amount of artisan crafting page/tome depending on your level, with a small chance of dropping a recipe ranging from a 4 property item to legendary item to jewelcrafter recipe.
It goes on...
Edited due to Polaris's suggestions.