Nick Pittson wrote:
It like nodding your head when you make eye contact with someone: a sign of politeness and courtesy/respect (as Don mentioned).
It depends where you are, I suppose. If you cross yourself when you enter a Catholic Church are you worshipping? Strictly speaking, I suppose that it depends upon what your inner mindset, but at the very least you are performing a ritual of worship. Some dojo have everyone bow to a shrine and/or perform some kind of basic Shinto ritual, some don't.
Now depending upon your religious beliefs, this may or may not be a problem. Despite what many Aikido people say about a gesture of respect, many people have problems with performing a religious ritual outside of their own religious beliefs - regardless of the mindset used when performing it. Some people's religious beliefs even include very specific limitations on what or who can be bowed to (again, regardless of the mindset when the bow occurs).
Japanese people tend to be fairly flexible about religion and religious practices. I've spoken to more than one Japanese shihan (all students of M. Ueshiba) about this, and the response was invariably along the lines of "If you don't want to bow than don't bow". I practiced at numerous dojo in Japan, and I can't imagine that kind of thing becoming a major problem - most Japanese would probably just look at it as another odd foreign quirk. Interestingly, the only people that I've run into that try to hold a really hard line about the bowing stuff were non-Japanese.