Embedded sensing can benefit soft robots with the ability to interact with their environment but producing embedded soft sensors can be challenging. Multi-material Fused Deposition Modeling (FDM) additive manufacturing allows producing complex structures, by combining more than one kind of polymeric material. For multi-material FDM, conductive thermoplastic elastomer filaments have been developed. This allows the printing of flexible functional structures, based on thermoplastic elastomer structures with conductive paths that are of great interest for stretchable electronics and soft robotic applications. In this study, stretchable piezoresistive elastomer strain sensor composites were successfully produced by using multi-material FDM. A piezoresistive thermoplastic elastomer was printed on the top of a nonconductive, flexible thermoplastic elastomer strip using FDM multi-material 3D printer. FDM elastomer filaments with different shore hardness as substrate materials for the gripper structure were used. The hardness of the elastomer affected the printability and the adhesion to the conductive elastomer material, which was used as a strain sensor material. The hardness affected the strain sensor properties too. The piezoresistive response, dynamic behavior, drift, relaxation and sensitivity of the printed multi-material strips were investigated by tensile tests. Soft robotic grippers with integrated sensing elements to detect deformation while touching the objective were selected as a case study. The soft grippers with the integrated sensors exhibited intelligent response by recognizing when they were griping a small or big object and when an obstacle was inhibiting their function.