A high-priority target in the design of new metallic materials for load-bearing implant applications is the reduction of Young's modulus approximating that of cortical bone in the predominant loading direction. Here, we explore how directionally preferential bulk elastic properties of implant materials are achieved by harnessing elastic anisotropy. Specifically focusing on recently proposed biocompatible refractory high-entropy alloys (RHEAs) in the body-centered cubic structure, we conduct systematic densityfunctional theory calculations to investigate the single-crystal elastic properties of 21 Ti-containing RHEAs. Our results provide evidence that the valence electron count has a dominant influence on elastic anisotropy and crystal directions of low Young's modulus and high torsion modulus in the RHEAs. By means of modeling the orientation distribution function for crystallographic texture, we examine the effect of non-random texture on the anisotropic poly-crystalline Young's modulus and torsion modulus with varying texture sharpness. We adopt fiber textures that can result from rolling and distinct texture orientations that can form during rapid directional solidification. We discuss the potential for lowering Young's modulus in the RHEAs by using single crystals or textured aggregates. Furthermore, we prepare four of the theoretically considered alloys by arc-melting and report their lattice parameters, quasi isotropic Young's moduli, and Wickers hardnesses.