The Torah states that some of the wool tzitzis strings should be dyed with techeles dye. Techeles is a dye made from the blood of the chilozon. There are various opinions among the Rishonim as to how many strings should be dyed. Mishnah Berurah quotes the view that two strings on each corner should be dyed blue. Min haTorah, one should even affix dyed woolen strings to a linen garment, since the mitzvah of wearing techeles overrides the aveirah of wearing shatnez. Nowadays, since we no longer have techeles and the mitzvah of tzizis is no longer kept in its entirety, it is forbidden to affix woolen strings to a linen garment. It is certainly forbidden to attach linen strings to a woolen garment. As a safeguard against shatnez, it is advisable not to wear garments or strings made from linen at all.
[שו"ע ט, ב ו־ו, ומשנ"ב ז, ח ו־יז; וראה שעה"צ כז; וראה ביאורים ומוספים דרשו, 21]