Compact quantum groups can be studied by investigating their co-representation categories in analogy to the Schur-Weyl/Tannaka-Krein approach. For the special class of (unitary) "easy" quantum groups these categories arise from a combinatorial structure: Rows of two-colored points form the objects, partitions of two such rows the morphisms; vertical/horizontal concatenation and reflection give composition, monoidal product and involution. Of the four possible classes $\mathcal{O}$, $\mathcal{B}$, $\mathcal{S}$ and $\mathcal{H}$ of such categories (inspired respectively by the classical orthogonal, bistochastic, symmetric and hyperoctahedral groups) we treat the first three -- the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations of block size, coloring and non-crossing conditions. This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. The article is purely combinatorial in nature; The quantum group aspects are left out.