I've only just started learning about the Czech Republic recently because I found out I had ancestors from there. I know it spent some time called Bohemia, and it also spent some time being owned by different groups (Duchy of Bohemia in the Moravian Empire, Kingdom of Bohemia in the Holy Roman Empire, etc) and I'm trying my best to learn about it. Wikipedia seems to talk about what names it took as it switched hands or came into its own, but none of the Wiki articles seem to go into why.
First off, you asking about more than a thousand years of history - that's rather broad question, if you want some comprehensive answers, you'll have to specify it a bit. Also, as was already said, your premise is wrong. Bohemia is actually one of the more stable European political entities, being united as one nation under one ruler since 7th century or so. Sure, ruling dynasties changed, and we were for four centuries subjected to Habsburgs, but really, just read up on some neighboring nations, you will see that's simply how history was. And now for your question: Heritage. There was a problem after Přemyslid dynasty died out in 1306 and Habsburgs were trying to get their hands on Czech throne. Czech nobility rather choose Jan Lucemburský (John the Blind) but same thing happened later in 1437 with death of Zikmund Lucemburský (Sigismund of Luxemburg) - that was further complicated by Hussite wars. Next years were indeed rather wild (I would suggest reading up about the leading figures of that time, as Hussite times are probably the most interesting part of our history), with Czech nobility trying all in their power to keep Habsburgs out and searching for suitable candidate all over, and ultimately failing. After that, Habsburgs ruled until Great War. There's certainly more to it then this, but, yeah, basically it was about a guy not having a son.
Could you maybe expand on that a bit? Obviously there have been a couple of ruling dynasties but I don't really see the Czech lands "changing hands" that much... you basically got Bohemia proper for about six centuries, then the first major shift with Bohemia in the so called Habsburg Monarchy for another four centuries or so, then another with Bohemia->Czechoslovakia for about another century. That hardly seems like "many times" to me.