I am no expert and do not pretend to be, but I will offer what knowledge (hopefully correct) I have gained from my readings.
First of all, I think you are wrong in your assumption about ancient Persian (Achaemenid or Arsacid) armies and the like. While they are often portrayed as poor (at best) soldiers who rely on mass manpower, during many times in the history the Persians were exceptional warriors with lots of exceptional warriors under their command (they were an empire that incorporated a vast array of cultures and people, so their armies were always very diverse with specialized troops who were used to fighting in a variety of terrain types in a variety of styles).
The Achaemenid suffered defeats at the hands of the Greeks because of a number of different factors.
1st of all, and most importantly, often times their soldiers were conquered people without loyalty or a reason to fight for their emporer. If they were more afraid of their enemy than the consequences of disobeying, they would run. (and who could blame them?)
2nd of all, they were fighting exceptional, world class warriors in Greece (a description that would also fit many Persian troops) who were fighting to defend their homes and families. That is a powerful motivation and probably contributed a lot to the Greek's success.
3rd of all, it is true that Greek weapon and armour technology was certainly better than what average Persian soldiers would use.
4th of all, the Greeks had a style of fighting that was almost specifically designed to counter the way many Persians fought. Their equipment, style, formations, tactics, etc were honed after centuries of defending their homeland against horse-mounted foes loaded to the teeth with missiles (there was much more to many Persian armies than that, but they were still an essential factor). More importantly it was a style that was designed for and well suited to the type of terrain in Greece (and much of Anatolia where the Greeks made great inroads).
5th of all, Persians had to try to control a massive Empire rank with rebellions. Often times loyalty was an issue with troops as much as morale.
The Greeks actually had a lot of respect for many people fighting in the Persian army, so I doubt that they were bad troops.
Also, you mention Romans, look what happened when Romans fought the Arsacid Empire (now this I do know a lot more about). The Romans always performed well on the own terrain (wooded, hilly, etc) they were used to fighting in, and the Arsacids almost always outperformed the Romans in the terrain they were used to fighting in (vast open areas of land), despite having serious disadvantages in manpower and likely wealth.
The thing is that both the Roman and Arsacid way of fighting dominated in their respective Empires, but failed to perform well outside of it. What works well one place will not always work well in another, and both militaries were products of their national experiences.
Don't forget though, when they chose to use it, the Arsacids had some pretty effective infantry at their command (the type that performed well in hilly terrain). I remember reading about how a force of Median infantry (Media was part of the Arsacid Empire at this point) wiped out a Roman legion (and if I remember correctly, they were also at a disadvantage of numbers).
I wouldn't be so quick to assume that the ancient Persians were bad warriors. The Medieval Middle East though, based on what I have read, I would need to agree with you for the most part. They seemed to be pretty lousy. I am guessing again though that that had something to do with morale of troops who were only fighting so that their insane Caliph did not have the testicles chopped off (yes, exaggeration for effect).
Byzantines on the other hand were fighting for their homeland. Lot's of Crusaders truly believed in the Religious reasoning for the Crusades and were fighting to reclaim the Holy Land from unbelievers and stop the persecution of pilgrims. Those are a lot more powerful motivations than fear of a whip.
Also, I believe that the West had more of a warrior culture, whereas in many areas of the Middle East, that warrior culture did not exist and people lived in fear under an authoritarian rule.
Maybe I am wrong, but that is the impression that my readings have given me. If you want proof, look at the Mongols and what happened to them when they settled in the Middle East and adopted Middle Eastern culture and traditions.
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