Japanese people started to have some nuts and fruits with Japanese green tea many years ago. They are called Ocha-uke. Ocha uke is taken for not only emphasising a tea flavour but also protecting your stomach from Catechin and Caffeine. If it is not an official tea ceremony, you can choose any kind of sweets for Ocha-uke. These are some examples of best combinations of tea and Ocha-uke (sweets).
I enjoy Sencha accompanied with shortbread.