I think it is to do with having two pistons changing direction at the same time. With an even number of cylinders, they do, which would produce a more severe vibration. With pistons as large as the ones they used, this would have been a significant factor. This is the principle behind five cylinder car engines (Volvo et. al.).
I’m straining my brain on this one. I attended mechanic school in 1949. What is quoted above is quite possibly the reason but the internal geometry of the engine may also come into play. On a radial engine the connecting rods are not quite the same as on a conventional internal combustion engine. On an auto engine the crank throw passes through a 360-degree path and the connecting rods pass through the same path. On a radial engine there is what is called a master rod and this is attached to the crankshaft which is similar to that of an auto engine. The center of the master rod passes through a 360-degree path. The other rods are called articulating rods and they are attached to the master rod at the periphery of the master rod and as such when the master rod passes through a circular path the articulating rods pass through an elliptical path. I believe this type of set-up can not accommodate an odd number of articulating rods. (The master and the articulating rods total up to an uneven number).
Because the articulating rods pass through an elliptical path the firing of the spark plugs must be altered so that when the engine passes through two revolutions all of the plugs fire at the same position of the piston. This is accomplished by the use of a compensated cam on the magnetos, which is profiled to cause the points on the mags to open at the correct time. The cam rings on the engine are profiled so that the valves open and close at the same time to compensate for the uneven movement of the pistons.
At least I think this is correct.
Check this out:
http://www.howstuffworks.com/radial-engine2.htm
