We present a comprehensive computational study of sp3-carbon allotropes based on the topologies proposed for zeolites. From ≈600 000 zeolite nets we identified six new allotropes, lying by at most 0.12 eV per atom above diamond. The analysis of cages in the allotropes has revealed close structural relations to diamond and lonsdaleite phases. Besides the energetic and mechanical stability of new allotropes, three of them show band gaps by ca. 1 eV larger than that of diamond, and therefore represent an interesting technological target as hard and transparent materials. A structural relation of new allotropes to continuous random networks is pointed out and possible engineering from diamond thin films and graphene is suggested.