I’m not going to act like I am a scholar of American history. I am not. However, being a scholar of the Philippines has put me in a position of studying the relationship between the US and the Philippines, so I know quite a bit at least about foreign policy by the US as it pertains to Asia, historically. So I feel like I want to defend Zinn a bit here. He wrote that book to attack some of the popular historiography or historical viewpoints of the time, and how Americans are taught their own history. Very rarely, for example, do Americans learn about the unwarranted and unneeded invasion of the Philippines, an invasion that resulted in perhaps millions of Filipino deaths. Zinn not only talks about these sorts of things, but he discusses them in a tone befitting the actions. He goes to great lengths to tear down certain types of narratives that erased the ethical and moral aspects of history.

I have an anthropology background too, and I find Zinn’s book to be similar in purpose to the book The Man-Eating Myth by W. Arens. Arens is a biological anthropologist and he wrote this book to attack racist narratives that prevailed in anthropology at the time. Does Arens’ book still hold up completely today in terms of his data? No, it doesn’t. Some of his assertions were later discredited by thorough research. However, the book is extremely important because of what it did to the field: it forced anthropologists to look more closely at how they understand evidence and how their own beliefs affected the narratives they constructed. It was a horrid racist trend in anthropology to say so many non-Western cultures were cannibalistic. The evidence was shoddy and Arens rightly attacked it. The book is still important now for that reason.

I feel similar for Zinn. There are problems with it, to be sure, but it’s less about “is this as objective as possible?” and more about “this is trying to reject bad trends in history”. I have read Jill Lepore’s book and I think it’s quite excellent, by the way. I would strongly recommend it.

I would love other historians to discuss this here honestly.

I recommend you read this comment from /u/Freedmenspatrol that addresses some specific questions about the book, while also touching on more general criticisms.

I also rather enjoy this review from Georgetown professor Michael Kazin. Kazin is genuinely frustrated with the way Zinn dismisses the motivations of those he disagrees with, attributing nearly everything to manipulation and deception. This may be a People's History, but those people are constantly being duped and defrauded to the extent that you have to wonder if Zinn thinks much of them at all:

Zinn’s big book is quite unworthy of such fame and influence. A People’s History is bad history, albeit gilded with virtuous intentions. Zinn reduces the past to a Manichean fable and makes no serious attempt to address the biggest question a leftist can ask about U.S. history: why have most Americans accepted the legitimacy of the capitalist republic in which they live?

Kazin also criticizes Zinn for leaving out religion. What does that look like? Well:

After a remark about inter-religious motives for urban violence in the 1840s, "religious" only appears 5 more times in the book- that's 412 pages of my PDF version. In fact, the word "Christian" only appears 16 times in all 622 pages; "Catholic" only 19, "Methodist" and "Baptist" only twice, "evangelical" only once. "Jewish" appears 12 times and "Jews" 13 times, but if we take out references to Jewish people in other countries, organization names, and where they're an example of an immigrant group, that leaves just two instances. A handful of other Protestant sects are name-dropped exactly once when they opposed a war, Islam, Hinduism, Orthodoxy, and Buddhism are entirely absent.

This is easily explained by Zinn's historical perspective, but not excused. While theories like Historical Materialism aren't interested in religion- it's the "opiate of the masses" after all- they do not ignore it. They offer idiosyncratic interpretations of its function and nature. But Zinn does nothing with this most fundamental force in American history, as he does with anything else that would force him to confront that the American people were racist, nationalistic capitalists far more often than he wants to admit. Racism in particular is either presented as a political tool or like a weather forecast: "Racism was strong" or "It was a time of intense racism in the United States." The way it integrated itself into the minds of American people is hardly addressed. Yes, Zinn has chosen a particular topic to write about here, but that's not a choice you can make freely. You have to prove that that topic can stand on its own, and Zinn does not. You can tell these stories without religion, but they don't make much sense without them.

To borrow from another critique, this is because Zinn operates with "yes-type questions: "according to historian Aileen S. Kraditor, yes-type questions send the historian into the past armed with a wish list [...] 'If one historian asks, "Do the sources provide evidence of militant struggles among workers and slaves?" the sources will reply "Certainly." American history is 250 years of millions of people; you can write a history of anything and provide ample evidence. Did people think the US should be a monarchy? Yes, and I can provide plenty of primary sources to prove it. That's a bad take though, because it's not representative. This is the metric by which Zinn's history fails. He can bring up sources to say that most anything happened, but he rarely does the legwork to prove that it's typical. Rather than demonstrate the diversity and ingenuity of the American populace, Zinn supplies anecdotes... and more anecdotes.

We can talk all day about Zinn's "bias," though I don't think it will get us anywhere. Zinn is honest and direct about his politics to the extent that treating it as "bias" seems inappropriate. Those politics include goals which most historians would agree with: textbooks are generally too nationalist, they focus too much on elites, concerns of class and race are under-represented. I am glad that Zinn's book got people thinking about this. He covers more of the late 19th and early 20th century labors issues than many popular authors or school textbooks, and the resulting discussions of class are necessary, if not well supported by history.

Nevertheless, it is not a good introduction on how to do history. Zinn's citations are absent or muddled in a single-page "bibliography." He is more concerned with telling the right or true history than with understanding the motivations, goals, and strategies of past peoples (a common fault of these types of "textbook alternatives". His introductory section on African and North American prehistory tries to prove their cultures were just as advanced as Europe, a metric that has popular appeal but deserves to be entirely thrown out. Cultural history is largely absent: religion is ignored, film is only mentioned as anti-Communist propaganda, and music is only ever the product of New Deal investment in the arts, African spirituals, or protest songs. The result is a book that mentions "war" at least 970 times and whose focus on the wars, business, and politics of American history make it feel a lot closer to standard curriculum material than it would like to admit.

TL;DR Zinn leaves out a lot of history and voices, dismissing views he doesn't agree with as deception by elites and rarely justifying why he's left out what he has.

More can always be said, but while you wait for new answers, you may wish to read the FAQ section on Zinn's work.