I just find cash a bit cumbersome, probably because I'm used to using my debit card for everything, and something requiring cash payment ends up being a hassle. But yeah, cash makes my wallet too bulky to fit comfortably into pockets, it requires actually having to withdraw (sorry for the deposit, I always mix those up) money beforehand (why not just pay with the card in the store instead?) and it takes longer at the store because the cashier has to count the money. And with decent bank deals, where you don't get fees for payments, it's a pretty sweet deal. Sure, there are security concerns, but I'm careful and the banks cover most losses anyways.
Pretty much the only stuff I pay in cash unless I for some reason have received cash from someone else, is the occasional bus fare when my bus card is expired or soda from soda machines :/
ETA: Which sites only accept credit cards? I have never run into that problem. I know banks here won't give people under 18 a normal Visa (debit), but they'll get a Visa Electron, which doesn't allow for online shopping, but which still works as a debit card for regular stores and in a lot of foreign places too.