So I get to be the first to resort to Wikipedia - which, in this case, makes as much sense as anything else. I hadn't thought about their last point (that people stopped using inline engines by the end of WWII), but I think it's true.
Comparison with inline engines
ProsWeight: Air-cooled radial engines often weigh less than equivalent liquid-cooled inline engines.
Damage tolerance: Liquid cooling systems are generally more vulnerable to battle damage. Minor shrapnel damage easily results in a loss of coolant and consequent engine seizure, while an air-cooled radial might be largely unaffected by minor damage.
Simplicity: Radials have shorter and stiffer crankshafts, a single bank radial needing only two crankshaft bearings as opposed to the seven required for a liquid-cooled six-cylinder inline engine of similar stiffness.
Reliability:The shorter crankshaft also produces less vibration and hence higher reliability through reduced wear and fatigue.
Smooth running: It is typically easier to achieve smooth running with a radial engine.
Cons
Cooling: While a single bank radial permits all cylinders to be cooled equally, the same is not true for multi-row engines where the rear cylinders can be affected by the heat coming off the front row, and air flow being masked.
Drag: Having the cylinders exposed to the airflow increases drag considerably. The answer was the addition of specially designed cowlings with baffles to force the air between the cylinders. The first effective drag reducing cowling that didn't impair engine cooling was the British Townend ring or "drag ring" which formed a narrow band around the engine covering the cylinder heads, reducing drag. The National Advisory Committee for Aeronautics studied the problem, developing the NACA cowling which further reduced drag and improved cooling. Nearly all aircraft radial engines since have used NACA-type cowlings. Because radial engines are often wider than similar inlines or vees, it is more difficult to design an aircraft to minimize cross sectional area, a major cause of drag, although by the beginning of the Second World War, this disadvantage had largely disappeared as aircraft sizes increased, and multi-row radials increased the power produced in relation to the cross sectional area.
Power: Because each cylinder on a radial engine has its own head, it is impractical to use a multivalve valvetrain on a radial engine. Therefore, almost all radial engines use a two valve pushrod-type valvetrain which may result in less power for a given displacement than multi-valve inline engines. The limitations of the poppet valve were largely overcome by the development of the sleeve valve, but at the cost of increased complexity, maintenance costs and reduced reliability.
Visibility: Pilot visibility may be poorer due to the width of the engine on single-engine aircraft, although tight fitting cowlings helped reduce this factor somewhat. Equivalent inline engines often resulted in overly long noses, which similarly impaired visibility directly forward.
Installation: It is more difficult to ensure adequate cooling air in a buried engine installation or with pusher configurations.
Size: The smallest classes of radial engines, with three and five cylinders are very rough running and unreliable when compared to equivalent four cylinder inline or horizontally opposed engines which later became more popular for light aircraft as a result.
While inline liquid-cooled engines continued to be common in new designs until late in World War II, radial engines dominated afterwards until overtaken by jet engines, with the late-war Hawker Sea Fury and Grumman Bearcat, two of the fastest production piston-engined aircraft ever built, using radial engines.
https://en.wikipedia.org/wiki/Radial_engine
