Are you required to wear a four conered article of clothing in order to satisfy the mitzvah of Tzitzis?
The strict mitzva of Tzitzis is "put fringes on your four-cornered garment." Biblically, it was the norm that people's clothes were four-cornered.
Today, modern fashion has generally done away with four-cornered garments, and thus the technical requirement; nonetheless, we try not to get out of mitzva obligations, so standard practice has become to wear a four-cornered garment with Tzitzis on it.
Between the easy availability of the mitzva; the fact that it has become normative; and various statements in classical sources about the positive effects of wearing tzitzis (and negative effects of avoiding them), it is appropriate ("very strongly should do") to wear a four-cornered garment.
I'm not sure how people interpret the word "required", so I'll leave it at that.
The Gemara in Shabbos (Lamed beis,amud beis) says because of the Sin of not wearing Tzitzis Children die but the Tosfos there explains that it is talking in their times when the regularly wore clothing of four corners and did not place Tzitzis on them but today we do not wear such clothing so no responsibility exsists but the Tosfos ends it is good to buy such an article of clothing and to make a Bracha every day. Reb Moshe Feinstein in the Iggres Moshe was asked this question by his son and he answered straight hallachicialy there is no requirement.But since all of Klal Yisroel accepted the Minhag it is Assur do do otherwise as is with any other Minhag Yisroel based on the Possuk "SHEMAH BNI MUSSAR AVICHACA VAL TITOSH TORAS IMECHA" Plus he adds we know the TZITZIS is counted as if you did all mitzvos and Reb Shimon Bar Yochaisays you will be Zocheh to see the Shechinah and it will save you from sin and the many other Things mentioned in Shas in Praise of those who are carefull about Tzitzis. In conclusion according to Reb Moshe it is Required today to wear a four cornered article of clothing in order to satisfy the mitzvah of Tzitzis.