I usually wouldn't. But it would be fine either way.
"The 1970s' music industry" means "The music industry belonging to the 1970s." That makes sense figuratively.
"The 1970s music industry" means "The music industry of the 1970s." That's more direct and literal, and I think it's the way most people here in the U.S. would usually write it.
It also depends on the context. So all in all, I wouldn't worry about it.
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