I’ll second Chihayafuru, and add Hikaru no Go (different game, very shounen anime, still manages a surprisingly good emotional payoff if you stick with the show long enough).

I’m going to add a couple of more dubious ones: Princess Tutu, to the extent that it’s about ballet and not fairytales. And the old series Ayatsuri Sakon, which is primarily a mystery series, but often uses backdrops that relate strongly to Japanese culture (mostly bunraku, since the main character is an apprentice bunraku puppeteer, but there’s also one arc that deals heavily with Noh theatre)—to be honest, those were more interesting than the whodunnit bits.