We present a new sampling method for the multi-scale design of polycrystalline materials, which improves the computational time efficiency compared to the existing computational approaches. The solution strategy aims to find microstructure designs that optimize component-scale mechanical properties. The microstructure is represented with a probabilistic texture descriptor that quantifies the volume fractions of different crystallographic orientations. However, the original microstructure design space is high-dimensional and thus optimization in this domain is not favorable. Instead, we generate property closures, which are the reduced spaces of volume-averaged material properties that are computed in terms of the microstructural texture descriptors. We observe that the traditional design approaches which are based on sampling in the original microstructure space and sampling on the property closure are inefficient as they lead to highly concentrated design samples in the solution space. Therefore, we introduce a new sampling method in the property closure, which creates simplexes using the triangulation of the property hull and then generating samples for each simplex. Example problems include the optimization of Galfenol and α-titanium microstructures to improve non-linear material properties. The new sampling approach is shown to obtain better solutions while decreasing the required computational time compared to the previous microstructure design methods.