So I just got back from a trip to Japan, and naturally the Sengoku Jidai really piqued my interest. I learned, however, that before this period there were another series of wars fought between the four great clans:
The Taira clan
The Fujiwara clan
The Tachibana clan
The Minamoto clan
The question is: the impression I get is that these clans despised each other, so why do we not hear about their actions during the Sengoku Jidai, when the country was again at war? Did they all die out?
Those were the major noble families of the early Heian period but they were pretty different from the samurai clans that would arise later.
You're probably thinking of the Genpei War (1180-1185) between the Taira and the Minamoto when you mention a series of wars between the four clans, but the Taira-Minamoto rivalry is really the only case of military rather than political competition between them that I can think of.
Anyway, the short answer is that we hear about their actions during the Sengoku period all the time; we just don't realize it. Virtually every samurai clan you can think of was (or at least claimed to be) a branch of one of those families. As the various branches of these families gained holdings far from the capital, they adopted new names derived from these lands. The Takeda clan was a branch of the Minamoto, for example, but adopted the name of the village in Hitachi province that served as their initial stronghold.