The primary reason is actually pretty simple. Ancient coins were made by taking a piece of metal, not necessarily molten but malleable, and 'striking' it, that is, compressing it, usually with a hammer or something similar, between two dies that will imprint the design on the coin. Really old coins maintain a somewhat round shape because the natural outcome of something being compressed in that way is for it to spread out relatively evenly from the dies. More advanced techniques and standardization meant this relatively round shape eventually developed into a circle.

That said, there are plenty of examples of non-circular 'coins' with other regular shapes, such as octagons, squares, etc. Things like silver bars were also not that uncommon, and are certainly much easier to mass produce from molten silver in casts. On the other hand, coins didn't merely operate as instruments of trade, they were also instruments of propaganda, so striking was very important for governments and leaders.

If you're interested in the striking process, this page has a pretty good overview:

http://www.classicalcoins.com/page103.html