It seems strange to me that a peoples who fought primarily with shields and spears would have no armor for their legs and arms. They should at least have some leather instead of bare skin, right?
It would depend on the movie or period pieces you are referring to, some I have seen have had fairly decent armor and some have not.
However looking at art from the time it appears that fighting without armor on your arms and legs was fairly common. And there are reasons for this.
Armor is expensive. Now this is a gross generalization, but for large periods of Greek and Roman history the soldiers had to bring their own equipment to battle. If they only could afford a helmet and breastplate, then that's all they would have.
You have to remember that both the Greeks and Romans were fighting with large shields. This by their very nature means one of their arms will always be protected. The other arm should always be held behind that shield waiting for their moment to strike. And even in that case there is some evidence that the Greeks and Romans (should the soldier in question be able to afford it) could have some armor on their exposed forearm. I do not know how ubiquitous this was though.
Greaves (armor for their lower legs) was readily available, however again it was not as common as helmets and breastplates. Since those are the most important things to protect. Now, the reason why legs are not as heavily protected has to do with body mechanics, angles, and formation fighting. Truth be told, aiming for the legs is fairly difficult, especially with a big shield in the way. Your spear or sword starts at your shoulder, so to attack the enemy leg means the length of your weapon has to angle down over most of your opponents body, drastically decreasing your reach as opposed to your opponents reach. And unless you're careful opens you up to attack. It just isn't a priority. The exception seems to be cavalry, where having your legs protected in some way would have an obvious benefit should you ever engage in combat with someone standing next to you and your legs are exactly at their striking level.
So leather armor is not entirely a modern notion, but was nowhere near common as people seem to want to believe. Frankly, padded cloths are just cheaper and just as useful.
Pants just weren't as common a feature in the roman and greek world. It was seen as a barbaric style until fairly late in Roman history.