Elastic gridshells are efficient structures covering large spans with little material. They are fabricated using a network of straight members deformed into doubly curved shapes. Researchers and builders have used two families of remarkable curves to generate gridshells with straight planks: geodesic gridshells, where the planks follow the geodesic curves and lay along the tangent plane of the surface, and asymptotic gridshells, where the planks follow asymptotic lines and are orthogonal to the surface normal. This article proposes the concept of pseudogeodesic gridshells, where planks follow pseudo-geodesic curves of a surface. This article introduces a scaling argument showing that the structural efficiency of pseudo-geodesic gridshells may be twice as much as geodesic gridshells. This is illustrated by a parametric study that implements linear buckling analysis on various pseudo-geodesic gridshells.