It all depends. In some dojos they focus alot on the traditional aspects of budo training and then you automatically include the cultural aspects as well. The culture, ethics, laws and time of the samurai era all formed how bujutsu was trained. This is still present in todays budo training to a varying degree. In some places you barely notice it at all. But at others (noteably in extant koryu swordschools and non competition styles like aikido) the traditional ways are still practiced. They are a part of the technichal heriteage.
It does not mean that you are supose to convert into neo shintoism or konfuciuism or buddhism or anything like that but to understand the difference between faith (or religion) and tradition (and respect).
Respect the teacher and his teacher. Respect the founder of the art, and respect your fellow training partner who allow you to use them to learn. Respect the rules of the dojo. They are there to ensure your and everyone elses safety and well being. And they were evolved out of the cultural context when the training was very much about life and death, about swords and spears.
On the other hand, this is not something for everyone. Not everyone wants to do this kind of traditional training. But then there will be some budo/bujutsu traditions that will be outside of them. For these traditions the tradition and techniques are one and the same.
/J
