Lattice structures can present advantageous mechanical properties while remaining remarkably lightweight. Precise lattice design can however be tricky to set up with classical 3D modeling methods as it involves very fine details. Interestingly, natural porous structures can present such latticelike or membrane-like features which motivates to seek for more bio-inspired approaches to microstructure design. In this paper we present a novel method to grow lattice-like and membrane-like structures within an arbitrary shape and aligned along an oriented field. Our method relies on the use of a dedicated anisotropic Reaction-Diffusion system guided by an orthotropic diffusion tensor field. Assuming for instance the diffusion tensor to be related to the stress analysis of a given shape allows to generate emerging stripes patterns aligned along each one of the principal stress directions independently. A globally coherent mechanical model conforming to the initial shape boundary and infilled with oriented microstructures can therefore be synthesized. Further, we demonstrate the capability of this approach to handle other types of oriented fields such as obtained through optimization of material directions in scenarios with multiple load-cases. Our approach relies on spatially and temporally local operations allowing for efficient parallelization. This permits user-interaction and automated adaptation of the design, even for fine meshes over large volumes. For instance, a designer can locally erase or "draw" over the structure and let it regrow and adapt as well as enforce regions to be deliberately full or empty. The proposed approach yields smooth and conformal oriented anisotropic geometrical patterns. This is related to recent effort in the Structural Optimization community on the topic of optimized oriented infills and microstructure de-homogenization. One of the resulting designs is validated by means of a full scale general nonlinear analysis showcasing the advantageous properties of oriented microstructures for stability and robustness to buckling.