Trevor, is it the service or the architecture you're interested in? Your original post seems a bit vague on that. If it's the architecture, then I would suggest just waiting until the church is open for a public concert or other event, rather than a religious service. Or, perhaps, if you don't mind the morbid part, check your local newspaper's obituaries and attend a funeral service.
If you really want to attend a service just to see how it's done, be sure to check with your priest first. It might even be a good idea to take along a mature Orthodox friend.
I'm curious about a lot of places, too. I've never been in a mosque, a Mormon church, or a JW Kingdom Hall. But, somehow, I seem to survive by staying where I belong
! It would take more than mere curiosity for me to visit some locations.