What is the reason why wool talis katanim have black lines on them? Why don't talis katanim of other materials have them also?
The stripes remind us of the techelet, as was mentioned.
Some Poskim say the dye itself was black, although the fish was blue. See Rambam Hilchos Tzitzis 2:2 (cited below)
Black is a color of mourning, and we are sad that we don't have techelet anymore. (I heard this explanation, but can't cite a source for it.)
Why only wool? Because techelet was only used for wool strings. It is invalid for the mitzvah to dye another material with techelet. Rambam Hilchos Tzitzis 2:1.
כיצד צובעין תכלת של ציצית לוקחין הצמר ושורין אותו בסיד ואחר כך מכבסין אותו עד שיהיה נקי ומרתיחים אותו באהלא וכיוצא בו כדרך שהצבעין עושין כדי שיקלוט את העין ואח"כ מביאין דם חלזון והוא דג שדומה עינו לעין התכלת ודמו שחור כדיו ובים המלח הוא מצוי ונותנין את הדם ליורה ונותנין עמו סממנין כמו הקמוניא וכיוצא בהן כדרך שהצבעין עושין ומרתיחין אותו ונותנין בו הצמר עד שיעשה כעין רקיע וזו היא התכלת של ציצית.
Ashkenazim (those who study the kitzur shulchan aruch) agree that the black stripe design is to remind us of techalet. I posed this question to a member of my shul (Chabad) and got a totally different, yet 100% plausible answer. There are some people from the Chabad movement that claim that the black stripes are a tool to be used to ensure that you can make a bracha on tzitzit. Part of the requirement for the bracha is visual (and it seems that you can use the black stripe and your tzitzit strings to ensure your visability!)
Striped clothing is one of the striking characteristics of the Oriental male Jewish dress. This seems against the medieval principle of avoiding party-colored garments. It is not an invariable custom, but is frequent enough to deserve mention.
And according to http://www.shamash.org/lists/scj-faq/HTML/faq/11-01-03.html
Why do tallit typically have blue or black stripes? The reason why the tallis is striped is simply because that was the fashion in Greece and Rome. But this doesn't answer the question of why blue or black? Tzitzis are supposed to include a thread of blue wool in each tassle. Most believe we do not know the specific dye needed for the mitzvah. In memory of this dye, some adopted a custom to place a blue stripe on the garment itself. Others decided to add a black stripe of mourning for the lost element of the mitzvah. The black stripe gained popularity in Europe of the 15th through 19th centuries, when black-and-white clothing was more common for Jews in general. The blue stripe is now seeing a revival in the 20th and 21st centuries, but it's actually the older of the two customs. It just seems to us to be more modern. Sepharadic Jews believes the debate over what color is appropriate precludes wearing colored stripes, so they wear white stripes (or a different weave) on their talleisim. Maimonides was of the "same color as the garment" camp. For Baladi Yemenite Jewry (those Yemenite Jews that resisted the influx of Syrian customs), Maimonides is the final word on Jewish law. So, they do not wear a tallis of any particular color. One will often find an older, more traditional, Yemenite man wearing a rich blue or red tallis with matching strings. With or without stripes.