A famous result of Sands states that a class S of associative rings is semisimple if and only if S is regular, coinductive, and extensionally closed. Here,we investigate semisimple classes in a Kurosh-Amitsur radical theory for semirings. We show that such a class S is regular, K-coinductive, and K-extensionally closed. But a characterization of semisimple classes of semirings needs a fourth condition, namely that S is inverse semi-isomorphically closed. We also obtain other characterizations and results for semisimple classes and for subdirect products of semirings.