I didn't know passive voice was something to avoid when writing. Your writing reads very nicely for me.
Your first sentance has an active voice. Someone is doing something--in your case, "people" are growing
flax, and it's worded so that the verb "have been growing" is attached to the people.
If your first sentence were passive, it would read "flax has been grown by people for thousands of year." The "subject" is the flax in this sentence, rather than people (even though, in reality, it's the people doing the work). The active version, that you used, is "people have been growing flax for thousands of years." To make it active, you want the verb attached to the subject who's actually doing the thing.
According to this tutorial (
https://writing.wisc.edu/Handbook/CCS_activevoice.html), a tip to look for is the word "by"--if it's in the sentence, your sentence might very well be passive. You can see how, when I turned your first sentence into a passive one, the word "by" appeared.
Your other three sentences are the ones your program thinks are passive. (
The other three sentences are thought to be passive by the program=passive version of the other sentence)
"In ancient Egypt, planting times were governed by the seasonal floods of the Nile"=The Nile's seasonal floods governed the planting times of anchient Egypt. You can see the word "by" in your original sentence. The words following "by" are what your subject should actually be: "Seasonal floods of the Nile." I switched it to "Nile's seasonal floods" because it seemed to flow better. You could also say, "In ancient Egypt, the Nile's seasonal floods governed the planting times"
"In Western Europe, flax was planted in the spring, and yet, in the Himalayas and Alps, oral legend tells that flax was sewn in the fall, at the same time as barley" I don't think this one is passive, but I could be wrong...
"China, India, the Middle East, Japan, Africa, and later North America, all had different planting times" This doesn't seem passive to me, either, but I guess it could be rewritten as "There were different planting times in China, India...." But, that's more confusing
I like how you started each sentence with the countries/regions it was about. That made it easy to follow. The second sentance might be improved by becoming "active," and the last two sentences sound good as they are.
Here's how it'd look with the second sentence active:
"People have been growing flax for thousands of years, in many different parts of the world.
In ancient Egypt, the Nile's seasonal floods governed the planting times. In Western Europe, flax was planted in the spring
; and yet, in the Himalayas and Alps, oral legend tells that flax was sewn in the fall, at the same time as barley. China, India, the Middle East, Japan, Africa, and later North America, all had different planting times."
vs
"People have been growing flax for thousands of years, in many different parts of the world. In ancient Egypt, planting times were governed by the seasonal floods of the Nile. In Western Europe, flax was planted in the spring, and yet, in the Himalayas and Alps, oral legend tells that flax was sewn in the fall, at the same time as barley. China, India, the Middle East, Japan, Africa, and later North America, all had different planting times."
(I think there should be a semicolon in "In Western Europe, flax was planted in the spring
; and yet, in the Himalayas and Alps, oral legend tells that flax was sewn in the fall, at the same time as barley" since "In Western Europe, flax was planted in the spring" is a complete sentence; and " and yet, in the Himalayas and Alps, oral legend tells that flax was sewn in the fall, at the same time as barley." When two things are combined that could be complete sentances, a semicolon ";" or a dash "--" or a colon ":" should be used, rather than a comma ","....If I remember my college grammar right, LOL!)