The mitzvah of tzitzit demands that in order to don a four-cornered garment, one must place tzitzit on those corners. Although one is not obligated in this mitzvah unless he wears a four-cornered garment, Rambam, Hilchot Tzitzit 3:11, states that one should strive to wear four-cornered garments in order to fulfill the mitzvah of tzitzit. However, not all garments have the same value in their ability to allow one to fulfill the mitzvah of tzitzit.

Cotton Garments
The Gemara, Menachot 39b, quotes a dispute between Rav Nachman and Rava regarding which types of garments are included in the mitzvah of tzitzit. Rav Nachman maintains that only wool and linen garments are included in this mitzvah. One is also obligated to place tzitzit on other garments. However the fulfillment of the mitzvah in wearing these garments is only rabbinic in nature. Rava disagrees, and asserts that all garments may be used to fulfill the biblical mitzvah of tzitzit.

Shulchan Aruch, Orach Chaim 9:1, rules in accordance with Rav Nachman, and therefore one can only fulfill the biblical mitzvah with either a wool garment or a linen garment. However, Rama, ad loc., rules in accordance with Rava that one may fulfill the biblical mitzvah with any garment.

The most practical difference between the ruling of Shulchan Aruch and the ruling of Rama lies in the use of cotton garments to fulfill the mitzvah. According to Shulchan Aruch, one would only fulfill the mitzvah of tzitzit on a rabbinic level by wearing a four-cornered cotton garment containing tzitzit. However, according to Rama, there is a biblical fulfillment in donning such a garment.

Although Rama, the decisor for Ashkenazic Jewry, does rule that all garments are worthy of fulfilling the biblical mitzvah of tzitzit, there are varying practices whether one should try to satisfy all opinions by wearing a four-cornered garment made of wool. Mishna Berurah 9:5, writes that a yarei shamayim (a G-d fearing person) should fulfill the mitzvah of tzitzit with a wool garment, both on a talit katan, the garment worn the entire day, and the talit gadol, the garment worn for the morning prayer services. The Vilna Gaon (as recorded in Ma'aseh Rav, Hilchot Birchot HaShachar no. 17) as well as Chazon Ish (as recorded in Shoneh Halachot 9:1) personally wore a wool talit gadol, but a non-woolen talit katan. Rav Moshe Feinstein, Igrot Moshe, Orach Chaim 2:1, writes that although it is proper to wear a wool garment rather than a cotton garment, if one is particularly uncomfortable wearing a wool garment in the summertime, one may wear a cotton garment. Nevertheless, a ba'al nefesh (a very scrupulous person) who wears a wool garment even in the summertime is worthy of a blessing.

Leather and Synthetic Garments
There are certain garments that are completely exempt from the mitzvah of tzitzit. The Gemara, Menachot 40b, states that a garment made of leather is exempt from the mitzvah of tzitzit. Mishna Berurah 10:11, states that there is not even a rabbinic obligation to place tzitzit on a four-cornered leather garment.

While four-cornered leather garments are not currently popular, the question arises regarding synthetic garments such as polyester, rayon, and nylon, and whether they are suitable to fulfill the mitzvah of tzitzit. The question of synthetic materials revolves around the reason why leather is exempt from the mitzvah of tzitzit. Levush, Orach Chaim 10:4, states that the reason why leather is exempt is because it is not woven. Based on the comments of Levush, R. Tzvi Pesach Frank, Har Tzvi, Orach Chaim no. 9, states that synthetic garments that are woven are comparable to cotton garments. If they are not woven but rather produced as one unit, they are exempt from the mitzvah of tzitzit.

However, R. Moshe Feinstein, Igrot Moshe, Orach Chaim 2:1, asserts that reason why leather is exempt from the mitzvah of tzitzit is because weaving is not a necessary step in the production of a leather garment. Therefore, even if one were to weave thin leather strands and produce a garment, there would still be no obligation to place tzitzit on that garment. Similarly, since synthetic garments do not require weaving in their production, there is no fulfillment in placing tzitzit on them even if one were to weave them.

R. Eliezer Waldenberg, Tzitz Eliezer, 12:3, suggest that perhaps the reason why leather is exempt from the mitzvah of tzitzit is because it is naturally designed as a garment. However, regarding other garments, one is obligated to place tzitzit on them since they require weaving or some other method of production. Synthetic garments which require a production process are no different than any other garment. R. Waldenberg rules that one should place tzitzit on a four-cornered garment made of synthetic materials. However, he states that one should not recite a beracha on such a garment.

Linen Garments
As stated above, one is biblically obligated to place tzitzit on a four-cornered linen garment according to all opinions. However, the Gemara, Menachot 40a, states that the rabbis placed certain limitations on the use of linen garments for the mitzvah of tzitzit. According to Rashi, ad loc., the rabbis prohibited placing techelet on a linen garment. The reason is because techelet is not only unique in its color, but it must also be made of wool. While the Torah does allow a wool techelet string to be placed on a linen garment, this leniency only applies if there is a fulfillment of the mitzvah of tzitzit. However, if for whatever reason, there is no fulfillment of the mitzvah of tzitzit, one violates the prohibition of sha'atnez by wearing such a garment. Out of concern that one might wear such a garment without adhering to the many laws of tzitzit and techelet, the rabbis banned placing techelet on linen garments. Rabbeinu Tam, Shabbat 25b s.v. Sadin, disputes the opinion of Rashi and claims that the ban is not limited to techelet. The ban extends to the use of any linen garment, even if no techelet is placed on the garment.

Teshuvot HaRosh 2:8, claims that the common practice is to follow the opinion of Rabbeinu Tam and to disallow the use of all linen garments for the mitzvah of tzitzit. However, he notes that upon arriving in Spain he noticed that many people used linen to fulfill the mitzvah of tzitzit. He suggests that they might have been relying on the fact that there is no techelet, and perhaps even Rabbeinu Tam would agree that there is less of a concern.

Shulchan Aruch, Orach Chaim 9:6, cites the opinion of Rabbeinu Tam as normative. However, Rama ad loc., mentions the leniency of Teshuvot HaRosh that if only linen is available one may use it for tzitzit, as there is no techelet available. It should be noted, that nowadays there are many people who place techelet on their garments, and Teshuvot HaRosh's leniency may not be applicable. This would apply even to those who question the authenticity of modern day techelet, as the concern exists that by allowing linen garments, it may lead to someone who does use modern day techelet to violate the prohibition of sha'atnez.