In this paper we furnish mean value characterizations for subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution $\Gamma$. These characterizations are based on suitable average operators on the level sets of $\Gamma$. Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach, Reade and Saks. We analyze as well the notion of subharmonic function...