i recommend the book the art of not being governed by james scott.
also, you might check theanarchistlibrary.org and search the essays and articles there.
The !Kung people of South Africa, I guess would be considered "primitive." You can find interesting articles about them on libgen like Sharing, Talking, and Giving: Relief of Social Tensions Among !Kung Bushmen by Lorna Marshall. If I think of more, I'll try to remember and post the article names.
This answer is dipping heavily into my lay-interest in anthropology. Because of that I want to preface it by saying that, as fascinating as I find anthropology to be, it is a pretty awful field of scholarship, with a terrible history of serving the interests of settler and colonialist interests in subjugating the very people it studies. Many modern anthropologists try to avoid this, but it is a thing, and there is a deep and deserved skepticism of anthropology on the part of colonized people, not least because it most often takes the form of the colonizer using their research to define the lives of the colonized. Fuck that noise. Because of this, I suggest scoping out the following in addition to the stuff I mention below:
Many ancient cultures (and some current cultures) are organized along lines that are not centered on state. In fact most non-agricultural peoples would have what I consider to be non-state social formations (hunter-gatherers, many horticultural and herding societies, and even some agricultural societies), the state being a particular form of hegemonic social order (of course, different people define the state differently, here is a link to an old question that perhaps we should be revisiting: http://www.anarchy101.org/9913/can-someone-please-explain-the-state-in-simpler-terms.While there are lots of cultures that have lived without the state, that doesn't mean they were perfect or a model of what anarchists necessarily desire. Many still had/have social hierarchies that anarchists would reject. Many were or are patriarchal, or have very strict gender roles and norms (though there are also many that are less rigid in their gender roles and have less gendered social stratification than the majority of state societies). To my thinking, being anarchist is not just being against the state, or just the state and capitalism. There are many many more spooks we need to chase away.Regarding starting places for research, baa, fa, and human offered some really good suggestions. I would also suggest checking out Society Against the State by the late anarchist anthropologist Pierre Clastres, who studied the Guarani, Yanomami and Chulupi in South America.While I am less a fan, Harold Barclay's work might be of interest as well, specifically his books People Without Government: an Anarchist Anthropology, Culture and Anarchism, and The State. He is also both an anarchist and an anthropologist, and his anthropological work was primarily in North Africa (Egypt and, I believe, Tunisia).I would also highly suggest Stanley Diamond's book In Search of the Primitive, and, for an interesting take on state formation, Fredy Perlman's Against His-Story, Against Leviathan!
Aragorn! is absolutely correct that Against His-Story is not a work one would call historically accurate (perhaps that being part of the naming? I don't know), but yeah, as an anarchist myth of civilization, it is an apt description of the civilizing process - both the constant drive to consume more and more of the uncivilized (using Hobbe's metaphor of Leviathan, but applying to civilization as a whole and not just the State), and the way in which in in order to survive encounters with civilization, people must arm and armor themselves, thus reproducing the mechanisms of the Leviathan themselves. That last part I particularly like; the idea that in order to fight against the monster we become it ourselves. Incredibly hopeless, and incredibly accurate.It is definitely foundational to anti-civ anarchy, anarcho-primitivism, and more generally green anarchy. I highly suggest giving it a read, it is available online here at The Anarchist Library (theanarchistlibrary.org), or (I think) Little Black Cart (littleblackcart.com) distributes the book.