It is fairly certain that Mycenaeans made use of linen armour. I was wondering how much evidence there was for other BA cultures such as the Eyptians or Hittites using linen armour. I can't find anything.
There's the linen backing on Tut's scale armour I guess. The horse armour in Amarna Letter 22 is leather. I wish I knew what cloth Iron Age Egyptian quilted armour was made from. Edit: I just saw your other post, and checked my sources, and it looks like the only evidence is for linen.
As far as I know, there is no evidence of linen armor from Mesopotamia. The sources are pretty few and far between, though, so it may well have been used anyway. The sources we do have mention metal (or rarely leather) armor, or they do not specify the material of the armor. To the best of my knowledge, none mention cloth armor. All of the archaeological finds are metal, but then conditions in Mesopotamia generally do not favor the preservation of cloth.
Sometimes (especially in the early 1st Milennium), wide belts were worn, which were certainly usually fabric, though bronze examples have also been found (especially in Urartu). They could have been intended to serve as some kind of basic armor for the belly, though more likely it was simply fashion, as they were worn at all times, not only in combat.
The fact is that linen was never used as an armour in the ancient near east, as far as we know, for the early civilizations.
As Jamie stated, there were broad belts, but it was not only a matter of fashion, it was also a matter of making the armour look like the fashionable clothes. If one looks at the history of armours, one will notice how often they came to be decorated like civilian dresses; there are wonderful examples of such a kind of treatments in the Renaissance collections of the Louvre, or the greek muscled cuirasses as well.
Now about the armour : the broad belt has two main goals : 1) It protects your back, kidneys, and belly somehow,
2) it also helps to delay the weight of the armour from the shoulders to the hips of the warrior, thus dividing the effort and allowing more mobility.
Well, in any case, I have an article coming about that, for those who are interested, stau tuned.