The grid layout actually stemmed from the Roman Military Camp, or castra. The design was efficient at troop communication, movement, and defense. The city of Rome was built (and grew) before the military camp design, so it was not laid out in this fashion. Cities built later were sometimes designed in the castra fashion so as to be more defensible.

You can read more about the castra in Polybius' Histories which describes it in detail (along with the Roman military system). You can also google image search for some diagrams of what it would have looked like.