Ok, so being a (La)TeX fan (most days) I'm also kind of a typography freak and I feel dirty when answering questions and typing boring "quotes" instead of the “educated” ones.
There is software that could automatically do this in the text, for example SmartyPants from the very same guy who created MarkDown, without messing with the code entered in questions/answers. I've also recently stumbled with BetterType which also sounds like a reasonable alternative. Either of these do not only do quotes but also dashes and ellipsis.
(As a side note, being on a Mac, I've just discovered that it's possible to type key combinations of
Option [+ Shift] and
] to get all kinds of quotes. But still it would be nice to have this consistently applied over all answers and questions on the site.)