I think an allowance is important. Early on, it teaches them about money. A child with an allowance will quickly understand that in order to buy something, they have to save up for it. They'll learn that if they want that game, or they want that big toy, they can't spend their allowance all on little things like candy or trading cards. It may not teach the exact value of a dollar, but it definitely prepares them for things like budgeting and being responsible with their money.
Now I don't think kids should be paid for their chores--but that's not what I view an allowance as. It's kind of a half-empty, half-full thing. A kid does chores for the week and at the end of the week, receives money. They could be paid for work that they should be doing anyway or they could be rewarded for a job well done [for work that they should be doing anyway]. And as more and more responsibilities are added onto a child's "workload" (ie, at a very young age, they set the table; as they get older/taller, they can start unloading the dishwasher and putting dishes away where they can reach, etc.), more can be added to the allowance too. Show them that if they're old enough to take on new responsibilities around the house, they can also prove themselves more responsible with money. It's all about learning lessons, imo.
Sure, if you don't get an allowance and still have to do your chores, you'll understand that they're things that you just have to do (whether out of respect for the home your parents run or because you realize it's just a necessity). But when your parents buy you all of your things, when you hit the real world and realize things cost money and have no idea how to budget your funds very well, it's a pretty big step. :/
I actually think people don't learn the value of a dollar until they have their own job and/or pay their own bills. Allowance doesn't help that much with that. :/ That and once a kid gets a job, they shouldn't get an allowance. :/ I know people who have jobs and STILL get money on a weekly/monthly basis from their parents. So they have all this extra spending money and they'll never learn to save up or anything because why should they? They have disposable income! But for learning to budget/save up/not spend everything at once? An allowance is great to teach that skill early on.
it looks like i'm contradicting myself all over the place in this post. i need to come edit it later argh. xD;