I can't answer for the entire period so I'll answer what I know.

In ancient Greece at about the time of the Persian Wars, the Greek hoplites wore mostly bronze breastplates cuirasses.

However by the Peloponnesian War and especially by the Theban Hegemony the bronze cuirasses were increasingly phased out in favour of linen/leather.

J.K Anderson thinks it's for decreased weight as skirmishers began to play a larger role in warfare. Luke Sarson thinks it's for cost-saving argues that in Greece less and less people were able to afford the traditional bronze armor. I see no reason that it's not both, but I don't believe there's any written accounts for the reason. The linen armor continued to see widespread use throughout Alexander's reign and the successors states.

Around the time of the Theban hegemony was also developed the half-armor. It was supposedly invented by Jason of Pherae, and was supposed to be linen armor but missing protection on the back. It was supposed to force the wearer to fight (your back is unprotected so you better face the enemy) but it could again be a cost-saving measure. It's never as wide-spread though.

J. K. Anderson - Military Theory and Practice in the Age of Xenophon
Luke Sarson - The Reforms of Iphikrates