Why can't we tie the bitcoin mining process to some sort of useful calculation? Maybe some variant of folding @home or something? We just need some problem with a hard to calculate, but easy to verify 'right answer' right?
Cryptographic hashes are designed to have these exact properties. If we wanted to replace this with a "useful" calculation, you'd essentially need to create a new hashing algorithm that also had some practical use beyond hashing a set of data.
Given that creating good cryptographic hashing algorithms is hard enough, I suspect that creating one that is tied to a "useful" result is not feasible.