I've recently learned that Muslim women aren't supposed to pray when they're on their period and do not fast when on their period. I've never heard of anything similar in Judaism. Can women pray when on their period? Also are there fasts they are permitted to eat/drink on if they are on their period?
In general, the state of being a menstruant does not place limitations on religious obligations. My understanding was in line with this statement: " When she is niddah a woman must continue to do all of her normal religious duties, like blessings and prayers. She should continue learning even with mentioning God's name when learning the verses of the Tanakh. One should not be lax during this time. Also the woman who is niddah can go to synagogue. She may touch holy books and objects without restraint. (Although there are some ashkenazim who are stringent about these matters.)" The quote comes from here which cites Taharat Habayit Vol. 1 Chapter 12.
This document, though problematic on some levels, certainly summarizes (starting page 11) similar ideas permitting and then introduces some issues of concern. The medieval changes (most of which did not catch on as normative) are also mentioned here, under the "Other Stringencies" section.
Finally, a recent responsa seems to fall in line with this trend -- in general, yes, a woman may go to the synagogue but there is a tradition of a minor change in behavior.