Wafers are round for a couple of reasons. One, you have to grow a giant pure silicon crystalline ingot (or boule). The growth process is done by hanging a seed crystal into a vat of molten silicon, rotating it, and drawing it out slowly. This is repeated several times until you have a desired size. The process is actually pretty similar to how one makes candles.
The boule is machined down to a true cylinder to make handling and a lot easier. Typically, there's also a slight flattening down one side as a handling reference. Among other things, the slicing and polishing process follows, and in general, the ability to evenly polish the boule and the wafers is a lot better when the shape doesn't have any corners.
The other thing is that because of the linear velocity differences as the boule is grown, the crystalline structure eventually starts to weaken as you get further outward in radius. Depending on the angular velocity, there may be a real point where the crystals are too small to be of use. Anyway, that means that no matter what shape you cut the boule down to before slicing and polishing, you can't avoid the fact that images printed towards the outer parts of the wafer are more likely to have defects anyway, so in the end, a circle shape is still equally efficient.