It really depends on the city and the place. The Greek city of Miletus was designed on a very modern looking grid system. Obviously a city with such a rigid grid would come with planning.

There's also a wealth of writing on city planning/urban design from antiquity. Aristotle's Politics Book Two Part VIII and Book Seven Parts XI-XII talk about urban planning in Ancient Greece with particular reference to Miletus. Plato wrote about urban planning in Critias and in Republic Book Two.

/u/jigglysquishy points to Miletus as an example - it's actually the best place to start. Miletus' grid plan was created by a man called Hippodamos of Miletus - he lived in Miletus during the Classical era, and laid out the new city plan (the city had been sacked by Xerxes' armies in the Persian Wars) on a grid, to avoid the chaos and clutter of earlier cities.

Many later cities were initially built on a similar "Hippodamean" grid pattern - most notably the cities of Priene, and the various royal capitals, like Pella, Antioch, Seleuceia, and Alexandria. Later Roman colonies also employed a grid pattern, often a simple repurposing of a military fort (since they already had straight roads, a public square, and walls). The city of Timgad in North Africa is an archaeological darling, because the grid pattern of the original colony is clearly visible, and you can see the different phases of the city as it grew during the Antonine, Severan, and Christian eras.

However, it's often really hard to tell which cities had a grid pattern - ancient cities either tend to get abandoned (and are rarely preserved as well as in Timgad) or do so well that they remain inhabited and are buried under a modern city. Generally, however, the layout of the city is dependent on a variety of factors, including the period (Hellenistic and early Imperial colonies are typically on a very strict grid pattern, but high Imperial cities are a lot more improvised - pre-Hellenistic cities were not organised), the importance of the site (A royal capital like Seleuceia is more likely to appear orderly than somewhere like Ai Khanum, which was initially just a military and trading outpost that eventually became important under the Greco-Bactrian kingdom), the geography (Antioch was squeezed between the Orontes and the mountains to the East). A well-planned city was clearly an ideal towards which the Romans and Hellenistic Greeks strove, but like a lot of ideals, reality tends to get in the way.