Sometime during its history, the SCA decided to ban all obvious use of religion. This is in order to stop any kinds of religious persecution, etc. occuring. I believe this law occured because incidents showed some kind of ruling was required. While taking away some period aspects of medieval times, it takes away many of the bad aspects as well as the few good. (eg it's not fun to be on the wrong side of a crusader, a catholic in reformation england, etc). In general, a guideline is to not inflict religion on anyone else, and everyone will get along fine. So while a catholic monk persona is out in the SCA, a scholar who dresses in monk like clothes, but with no obvious religious symbols is generally accepted.
One of the most obvious repurcussions of this ruling is the ban on certain potentially offensive symbols (eg the swastika, star of david,... ) in SCA devices.
 Actually, the ban doesn't really extend this far. Many people adopt personas that are deeply religious, because their medieval counterpart would have been deeply religious. The problem is only with inflicting your religious beliefs (even if only "pretend" ones) on other people who might find that offensive. (And some people might be offended if you pretend to hold religious beliefs or positions you don't actually hold; this is the actual problem with monk or nun personas, assuming that actual monks or nuns are not likely to join the SCA.) Feel free to carry or use a rosary or other religious artifact; just don't rub other people's faces in it, and don't force them (or appear to force them) to be a part of your religious observance.