Reddit mentions of Gardens and Gardeners of the Ancient World: History, Myth and Archaeology

Well, I've got three books that cover that.

The most straightforward would be Wild Harvest. It focuses on the role of plants in ancient societies. It's fairly useful to the general reader.

Another one that's good for general reading is Gardens and Gardeners of the Ancient World which follows exactly the timeline you're looking for. Starts off with more ancient history and societies before moving to how those methods influenced later cultures. Goes up to the Byzantine. There's an in-depth academic review of it over at Bryan Mawr Classical Review if you'd like to know more.

Then, though it's more of a textbook, there's Economic Zooarchaeology which doesn't just look at agricultural practices of ancient humans, but herding and hunting too. Amazon seems to be low on that one, so you might have a better bet going directly to the publisher. Here's a review of it from The Prehistoric Society.