Though I've played a monk with the Vow of Poverty, I'm convinced that any class can pull it off. First, you have to consider that there's a huge heap of benefits that come from sticking to the Vow.
In later levels, you get +8/+6/+4/+2 to four of your ability scores, respectively; you get immediate armor class bonuses, useful for whatever class you might choose, and these increase over the career of your character; you receive a plethora of Exalted feats over time (though I've never been interested in more than a handful of them); you gain saving throw bonuses; you gain damage reduction; and you get Exalted Strike (which might be particularly cool when coupled with a Paladin's Smite Evil ability).
Two more things: (1) a cleric with the Vow of Poverty should be able to use a holy symbol, but it would be a simple version of that symbol. This might require DM adjudication. I think everyone just assumes that a holy symbol in D&D is made out of pure gold, but the ones you see in horror movies are often made out of two chopsticks held at right angles to each other. Why not allow a wooden or bone holy symbol?
(2) Just because you cannot own magic items doesn't mean you cannot use them. Reread the rules on the Vow of Poverty. Under specific circumstances, someone who's taken the Vow can use
some magic items: they just have to be offered the use by someone else. In other words, your cleric could use a Potion of Cure Light Wounds if the party fighter decided to share. Likewise, your cleric could use a Potion of Cure Light Wounds found in said fighter's backpack to revive that fighter if he'd been knocked unconscious.
A character with the Vow of Poverty could, likewise, carry a magic item on his/her person for a specific purpose, as long as it is not to his own benefit. To elaborate, if your party is trying to destroy a powerful magic artifact and they've reached the final chamber deep in the pit of the volcano, and everyone else in the party is dead or otherwise unable to carry the magic artifact to its final destruction, your character can do so without repercussions. Likewise, you could decide you want to take that +5 giantslayer spear as your share of party treasure. You can't
use it, but you could carry it back to your church and donate it to your order so that it might be used or sold or whatever else the church wants to do with it.
All of that said, there's a real problem with D&D that players find themselves thinking, "If I can't
have anything, then I can't
do anything!" This isn't true, and the rules for characters who accept Poverty reflect that. Still, there's some necessary role-playing involved, and your DM needs to take the Vow into account when designing/running adventures.
In my monk's case, he's been really tough and hard to beat. I do think monks benefit greatly from the Vow, since the ability score improvements go into Dex and Wis, first, meaning you've got one heck of a good AC bonus. The only character class that I really think would have a problem is the wizard, with his Item Creation Feats and general reliance on magic items to make him more powerful. However, even that class might make for an interesting character.
Give the Vow a try... I think you might really find that you like it. As a DM myself, I've found the issue of not having to track personal items and magic a real blessing!
