Personally, I hate to ask for feedback. People who ask for feedback seem like ebayers - I use ebay a lot and I'm bombarded with "please leave 5 star feedback" and yet no real "thank you for your purchase" or any customer care like that. I give them feedback, though, not because they asked but because of their products.
I do my best to package my products safely and gorgeously - this can help people to leave feedback, and I add a hand written note. But, in a case where a buyer hasn't left any feedback, I take my boyfriend's advice: Wait a while, and if you're pretty sure they've received it but haven't left any feedback, send them a message saying that you hope the item arrived safely, that you hope they like it, and thanking them again for purchasing. This will remind them that they bought from you. I think it's a default setting to have an email sent to you if you receive a convo on Etsy. In which case, some people may read it as an email, or may log in. Or, they may log in to respond, and when they log in, there's a little banner at the top saying that they haven't left feedback yet. Failing that, it might at least remind them that the feedback feature is there.
If you're desperate for feedback, though, you can ask. It's not against the rules or anything, we're all trying to run small businesses here, afterall. It's just the way I see it. I leave feedback as soon as I receive items, though, so no one's had to contact me yet :P and it is always a good idea to wait and not leave feedback for a buyer until they do for you. If they leave negative feedback and spin some sort of story and really dramatically blow up a small misunderstanding, at least you can give your side of the story politely in their feedback box.
And as for immediately leaving them feedback just because they payed for an item, well on Etsy the order doesn't generally even get placed until full payment is made anyway, unlike ebay. To leave feedback for that would be a waste of time. You're better off leaving a thank you note in the package for that. Paying for the item was necessary and doesn't prompt feedback. It has to be immediate payment, or no dice.
I think it's important to leave good feedback for buyers, too, especially if they don't have any. It can sometimes make them look suspicious. I had an order from someone the other day, and his location looked a bit dodgey to me. But, he payed straight away, of course, and I sent the item out, and he left wonderful feedback. I had to leave him feedback for the good of other sellers, and the same goes if the individual had been a scammer, or had made outrageous demands that he knew couldn't have been met. Leaving feedback for a buyer - especially if they are exclusively a buyer - is so important for other sellers. If they're dodgey, then they know who they're dealing with, and it's within every seller's rights to refuse to serve anyone that they think is trying to scam them.