Sorry to join the discussion so late into the game. I'm an Orthodox Jew, and have travelled Europe when I was younger with my family, and more recently on my own to Asia, having studied Japanese history and culture exstensively. I also have a continued interest in some branches of comparative religion.
With my family on trips, I've been to famous Catholic and Protestant churches, where we'd be respectful, remove head covering where need be, but never participate in any sort of service or ritual.
In Japan, China, Thailand, and Singapore I've been in many Buddhist temples and Shinto shrines (and one Hindu temple). I will remove my shoes when neccesary (I don't think I've ever had to do it for religious reasons in those places) and I try not to give them any money beyond admission costs (tho I did pick up a few souveniers in some places). I don't do any sort of bowing, kneeling, clapping (wakes spirits up in Shinto), or joss-stick burning. For me, observing (watching) ritual and being respectful is suitable behavior.
I was really upset when on one trip to Japan a friend who was a bit more irreverent (and also an Orthodox Jew) decided to bow down in jest at a Shinto shrine. I don't think anyone saw, but it was typical "Loud American" behavior.
I would probably still enter churches, but I don't think I would be comfortable attending a religious ceremony in a church, especially Catholic. There are arguments in Jewish law about Protestant churches and mosques not being considered "idol worship," but Catholicism would be a harder sell. I don't mean that out of any disrespect, its just how my belief system views others.
Now, all that being said, it would be even a harder sell to say Shinto shrines and certainly Buddhist temples aren't "idol worship," But I guess given that the histories of those religions and mine aren't as intertwined, I'm more comfortable wandering in to one of those when ceremonies are going on.
I hope this wasn't offensive, just trying to inform.