Tempura is probably the best known Japanese dish after sushi and sashimi. It should be relatively easy to make: cut the veggies, throw in batter, fry in hot oil, but what I have heard, it appears to be rather tricky to get the consistency and taste just right.
To be honest, at this point, I have not tried making tempura myself. However, I did look up tempura recipes, and one of them is now on the Washoku page for Japanese food.
Enjoy – and let me know how it turned out!