Not sure what you are meaning by "truly Japanese" but there are many restaurants in traditional buildings even in Tokyo.
Matsuya, a soba restaurant in Akihabara, has 133 years of history.
Iseya, a tempura (particularly tempura rice bowl, tendon) restaurant in Minowa, has 127 years of history.
Visiting these historical restaurants may be a "truly Japanese" dining experience :)
What exactly do you mean by 'Truly Japanese?'
Do you want like home cooking or like kaiseki ryori?
Any particular area of town?