Underneath their linen tunics or dhotis, most West Asian men probably wore a strip of linen cloth wrapped around their hips like modern underwear, with the end tucked in to hold it tight. Unlike ancient Egyptians, West Asian men generally don’t seem to have wrapped their cloth between their legs. Instead they wore it like a little skirt.
Women probably wore something similar, at least sometimes. Women sometimes also wore another strip of linen under their dresses, around their breasts, to hold them in place.
Learn by doing: get a long strip of cloth and try to wrap it into underwear out of it
More about West Asian clothing
Ancient Egyptian, Mesopotamian & Persian Costume, by Mary G. Houston (reprinted from 1954 edition). For professional costumers, with patterns.
Women’s Work: The First 20,000 Years : Women, Cloth, and Society in Early Times, by Elizabeth Wayland Barber (1995). Not for kids, but an interested high schooler could read it. Fascinating ideas about the way people made cloth in ancient times, and why it was that way.