The organization of Roman cities is quite varied and depends on when the city was established and if there was a prior settlement on that site. The style of organization I believe you are referring to is urban planning consisting of a strong north-south axial road and an east-west axial road with public centers and places of importance placed at the intersection of those two axial roads. I think the most well known example from antiquity that remains is Timgad.

This style of Roman urban planning is based on the layout of Roman military camps, called castra. The north-south road was called the cardo and the east-west road was the decumanus. This was standardized as Roman military units were generally required to "dig in" and build a camp any time they stopped marching. Those camps consisted of a ditch and rampart around their rigidly organized tents. In the camps, the command structure would be centrally locating; in towns based on this pattern this later became the forum.

These camps were constructed at the scale required for the military unit. Castrum built for a legion were longer lasting and often evolved into cities.

Rome itself is a tangle of streets that tell a history of slow organic growth that follows the contours of her seven hills and response to the Tiber. Conquered cities might be restructured during times of prosperity to create the forum and surrounding public spaces. This is a simple answer based off my architectural education.

If you would like more information on the history of architecture I could suggest my former professor's book Ideas of Order (Jacqueline Gargus) or A Global History of Architecture by Frank Ching, Mark Jarzombek, and Vikramaditya Prakash. I appreciate the Global History book for the less Eurocentric focus and organization by timeline.