Perhaps the ultimate expression of the efficient aviation piston engine was the turbocompound engine. I'm not going into the intricacies of these massive - and massively complicated - beasts. Just keep in mind that what killed them was cheap jet fuel and the higher power density of turbojets (and, later, turbofans) combined with the turbine's greater mechanical simplicity.
Modern turbofan jets are about as fuel efficient as the best piston-powered aircraft of the Fifties while being mechanically simpler and more reliable. Also, companies such as Rolls-Royce are making high-efficiency turboprops. However, if fuel prices climb even that might not be enough for items which need to be moved more quickly than ground or sea transport can provide, but not as quickly as the more expensive jets move.
Using modern design aids and materials - which are part of what made modern turbofans so efficient - it should be feasible to build large piston engines - radial or in-line - more efficient than even a modern turboprop in the appropriate speed range, and requiring roughly the same amount of maintenance. Whether there is a niche large enough to make this economically feasible is another matter.
An interesting bit of aviation technology trivia: During the heyday of the closely cowled piston aircraft engine companies actually experimented with adding afterburners to provide increased thrust during takeoff or emergencies. (The latter included "Somebody is shooting at me!")
