I am wondering if you could clarify the manner by which the property of being Jewish is passed from mother to children. My concern is whether in the situation where my family history is unknown, if I could still be under an obligation to follow the Jewish faith because many many generations ago, along the line of mothers, there was a Jewish member of my family? I have only been able to find answers online discussing mothers and children, when I am worried about many many generations back.

Only if the Jewish ancestor is your mother's mother's mother's... mother, would you be Jewish. If you have no proof as such, than in all probability, logic and Jewish law, say that you follow the majority of the population, and need not "worry" or suspect that you may be Jewish. In fact, it's much easier to be a gentile who needs only follow the 7 Noachide commandments, rather than Jews who must observe the 613 commandments, and God loves all of His creation.