Japan had coins that were not round. Traditionally, round Chinese coins with a hole in the middle were used in Japan, but in the early modern period when Japan was producing its own coins the larger denomination coins were oval, ovaloid, and rectangular, as can be seen here at the Japan currency museum. http://www.imes.boj.or.jp/cm/english/history/16C/
I have seen images of the larger denomination coins stored in treasure chests so it is likely that they were created in those shapes for ease of transportation, as in the manner of gold ingots.
I have seen (and possess) smaller coins that were used by the common people. These were round or oval with a hole in the middle through which a string could be strung.
Everyday Life in Traditional Japan by Charles Dunn has some great information about how these coins were made and used.
As a follow up question, is there any particular reason for the trend of having a face on one side of the coin? I've seen this on most coins, especially those from Europe.
Before Qin unified China and standardized coinage into the round coins with square holes that we are familiar today. You have various forms of coinage used by other states illustrated here
The knife forms are about 18cm in lengths and are made of bronze. They may actually serve a practical purpose. And at least in its early forms people use it as a household item with edge kept sharp.
The other forms on the left came from a ceremonial vessel used in Zhou dynasty. 鼎。It symbolizes state authority and power, but it was nearly as efficient to carry as the Qin coins, they are stringed in bundles of ten or a hundred. Which is also why all later dynasties continued the tradition.
As for why coins are round in the first place, it was to prevent fraud. Since coins were generally made of a precious metal that had value even in non-coin form, unscrupulous money handlers might be tempted to snip off a bit from the edge of each coin, rendering it less valuable. A round coin would reveal that sort of tampering better than any other shape, since it would have a flat edge where it was snipped.
There non-round coins have turned up a fair bit over the centuries but yes there are a lot of round coins.
So why?
Well the manufacturing process. Early coins where mostly made by crating a fairly precisely weighed nugget of out a precious metal (gold, silver, electrum) and then flattening the nugget before placing it between two dies and striking the upper die with a hammer.
Its the making of the nugget that produces the initial round shape. There are two ways of doing this. The most common appears to have been to weigh out a certain amount of precious metal in the form of powder of very small nuggets and once you reached the weight of the coin melt them into a single coin weight nugget. This nugget would naturally be roughly spherical and when fattened roughly circular.
Another approach is to poor old the molten metal onto a hot stone (although there is little evidence of this approach) until you had a puddle the right size and then letting it cool. This again would produce a round blank which could then be struck.
Sources? Philip De Jersey's Celtic Coinage in Britain is fairly lightweight but covers the manufacturing process quite well.
A related question: For well over a hundred years in the 17th-18th centuries, Sweden minted these giant "copperplate" coins (Image from the Royal Coin Cabinet museum) that were supposedly literally worth their weight in copper. But I'm not sure why we did this - why not, for example, stick to more practical copper ingots instead? Additionally, wouldn't the value of the copper itself diverge from the nominal, minted value eventually? As far as I know we were still minting regular old silver coins at the time, so what made copper so special as to receive this odd treatment?
