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.
In Guide for the Perplexed 3:47 Rambam writes:
All these cases of uncleanliness, viz., running issue of males or females, menstruations, leprosy, dead bodies of human beings, carcases of beasts and creeping things, and issue of semen, are sources of dirt and filth. We have thus shown that the above precepts are very useful in many respects. First, they keep us at a distance from dirty and filthy objects; secondly, they guard the Sanctuary; thirdly, they pay regard to an established custom (for the Sabeans submitted to very troublesome restrictions when unclean, as you will soon hear); fourthly, they lightened that burden for us; for we are not impeded through these laws in our ordinary occupations by the distinction the Law makes between that which is unclean and that which is clean. For this distinction applies only in reference to the Sanctuary and the holy objects connected with it: it does not apply to other cases. "She shall touch no hallowed thing, nor come into the Sanctuary" (Lev. xii. 4). Other persons [that do not intend to enter the Sanctuary or touch any holy thing], are not guilty of any sin if they remain unclean as long as they like, and eat, according to their pleasure, ordinary food that has been in contact with unclean things. But the practice of the Sabeans, even at present general in the East, among the few still left of the Magi, was to keep a menstruous woman in a house by herself, to burn that upon which she treads, and to consider as unclean every one that speaks with her: even if a wind passed over her and a clean person, the latter was unclean in the eyes of the Sabeans. See the difference between this practice and our rule, that "whatever services a wife generally does to her husband, she may do to him in her separation"; only cohabitation is prohibited during the days of her uncleanness.
(Friedlander translation, my emphasis)
He also specifically makes mention of the stricter Islamic customs with regard to menstruation in a letter to his son R. Abraham1:
It is interesting that some of their judges follow the same practice with regard to a menstruant woman as the people of Magi of the Ishmaelites who are prohibited from looking at her or her clothes; conversing with her or treading the same path she does during that period.
See my answer here for some related sources.
1. According to R. Yitzchak Sheilat (in his edition of Rambam's letters) this letter is a forgery not actually written by Rambam.
The Rema OC (88:1) records multiple customs regarding a woman who is a niddah (menstruating) praying.
יש שכתבו שאין לאשה נדה בימי ראייתה ליכנס לבית הכנסת או להתפלל או להזכיר השם או ליגע בספר (הגהות מיימוני פ"ד) וי"א שמותרת בכל וכן עיקר (רש"י הלכות נדה) אבל המנהג במדינות אלו כסברא הראשונה. - There are those who wrote that a menstruant woman in the days when she sees [blood] may not enter the synagogue or pray [the Amidah], or mention the Name, or touch the [Torah] scroll (Hagahot Maimuni ch. 4); and there are those who say that she is permitted in all [of those], and that is the correct opinion (Rashi, Laws of Niddah). But the practice in these lands [Ashkenaz] is according to the first opinion.
However, the Mishna Berurah (88:6-7) writes the custom is for women to pray nowadays even while in niddah status
(ו) או להתפלל וכו' - ובבנימין זאב סימן קנ"ג כתב שלא נהגו רק שלא לכנוס לבה"כ ולא לראות ס"ת וגם כשמתפללת אינה עומדת בפני חברותיה ומשום מנהג וכבוד עושין כן ולא משום איסור עכ"ל וכן הסכימו האחרונים דצריכה להתפלל בביתה ולברך כל הברכות ובפרט ברהמ"ז וקידוש שהוא מן התורה וע' במ"א: (ז) אבל המנהג וכו' - ובמדינותינו נוהגין היתר לעולם ומברכות ומתפללות ומ"מ לא יסתכלו בס"ת בשעה שמגביהים אותה להראות לעם [ח"א] עוד כתב שלא יכנסו לבית הקברות עד שיטבלו:
Some Poskim suggest (see note 30) this may be the reason why many women are lenient regarding praying every day, in addition to other reasons.