In my limited but enthusiast understanding of the classical era I think the linothorax came to be very popular with the later Greeks. From a video I saw on YouTube of a professor with some students testing it, it seemed to pretty sturdy and could withstand arrows at close range while being light and cooler on hot climates like the ones in southern European. So, why did the Romans, that were so good at adopting innovations of their enemies (or by the way any later infantry armies) didn't used it? Or did they?
The Etruscans, contemporaries of the early Romans, did use a linen breast-plate. See Livy 4.20, in which he calls it a 'thorax linteus' that most commentaries note as λινοθώραξ (linothorax):
and had himself read the inscription on the linen breast-plate, I have thought it would be almost sacrilege to rob Cossus of such a witness to his spoils as Caesar, the restorer of that very temple...
Other contemporaries like the Samnites also used linen armour (tunicae argentatis linteae candidae in Livy Book 9.40) and there was even a Linen Legion (legio linteata) of Samnites in Book 10.3. Even Hannibal during the Second Punic War had his soldiers clad in linen armour: "mos est defendere lino..." (Italicus' Punica)
As for the Romans, it appears that some contingents DID use linen armour. In Book 4, Italicus shows that soldiers from Falerii were clad in linen armor, a 'native product' of the Faliscan region. The use of linen armor among the Romans continued into the AD period: the Emperor Galba was said to wear 'loricam linteam' (linen cuirass) by Suetonius, and Arrian in the 2nd century AD said in his Tactica that experienced horsemen wore 'corselets of scales, linen, or horn'.
For more information on linen armor, and the linothorax in particular, read:
Reconstructing Ancient Linen Body Armor: Unraveling the Linothorax Mystery by Gregory S. Aldrete.