Our favorite kappo restaurant also is cash only but for regulars, he mails out statements. I think we wire the money into his bank account directly.
The neighborhood sushi restaurant, I forgot my cash clip, tried to pay by credit card. Couldn't. It was no big deal. Owner says, pay next time. Please don't bother to make a special trip back to pay. All we had to do was to sign the check.
Walking around with a wad of cash seems pretty normal thing. I know that the grocery store near our home prefers cash so if you pay by credit card, you don't get store "points". You can redeem points for pretty awesome gifts or cash receive cash back.