I do not have any credit cards, and I prefer to deal with cash as much as possible. When I am going out to buy something, I will take the money with me; I am OK with carrying coins. Even if anonymous payment is available, I will still prefer to deal with cash.
If I am alone, I will still pay in cash (I will make sure to take as much money with me as I need or more). If I am going with someone else (which is more common), usually they will want to pay with a credit card (although I can still pay directly with cash, e.g. in case they do not have a credit card either or if they want me to pay for my own food instead), and if they want me to pay them back then I will.
As someone who doesn't use cash much, I'm curious about what happens when you enjoy some service like a restaurant, where you don't know that they don't take cash until you're done and they are waiting for you to pay? If they only take credit cards, and all you have is cash, what then? Return your food from your stomach? Surely they'd rather take your cash than have you walk out without paying. So at the end of they day, they aren't really no-cash companies...