Umi Dai Suki by Yoko Imoto
A sweet and simple story about a family's day at the beach.
Tebukuro o Kaini by Niimi Nankichi, Illustrated by Kuroi Ken (also available in English Translation under the title "Buying Mittens")
A sweet folktale about a fox cub visiting a human town to buy mittens in winter.
Boku wa Ukki by Claudia Bielinsky
A simple tale of a little dog and his adventures during an entire day. Excellent for learning Japanese!
In English, with Japanese themes or storyline:
Suki's Kimono by Chieri Uegaki, Illustrated by Stephane Jorisch
One Leaf Rides the Wind by Celeste Davidson, Illustrated by Kathleen Hartung
Where Are You Going? To See My Friend! by Eric Carle and Kazuo Iwamura
And a few extra, because you know I'm a cheater:
A Place Where Sunflowers Grow by Amy Lee-Tai, Illustrated by Felicia Hoshino
Wabi Sabi by Mark Reibstein, Illustrated by Ed Young