We theoretically investigate three-dimensional (3D) layered crystals of alternating graphene and hBN layers with different symmetries. Depending on the hopping parameters between the graphene layers, we find that these synthetic 3D materials can feature semimetallic, gapped, or Weyl semimetal phases. Using first-principles calculations to parametrize the low-energy Hamiltonians we establish the most likely electronic phases. Our results demonstrate that 3D crystals stacked from individual 2D materials represent a synthetic materials class with emergent properties different from their constituents.