To not be concerned at all; to not care whatsoever (strong negation).
彼の過去については一切気にならない。
I don't care about his past at all.
仕事に支障がなければ細かいことは一切気にならない。
I don't mind small details at all as long as they don't affect the work.
見た目は一切気にならないと彼女は言った。
She said she doesn't care about appearances at all.