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Transition metal dichalcogenides (TMDC) are currently drawing significant interest from the scientific community as 2D materials that have intrinsically semiconducting bandgaps. One additional advantage of TMDCs for discovering and developing materials with novel electronic, electromechanical, or optoelectronic properties is that both layer composition and registry can be readily tailored. To understand how such tailoring can expand the range of properties, here we used density functional theory calculations to determine the electronic structure and piezoelectric properties of bilayer TMDC heterostructures based on MoX2 and WX2, where X can be S, Se, or Te. For identical layers with no misorientation with respect to one another, we find that the registry of the two layers can change the bandgap type (direct vs indirect), as well as its value (by ≈0.25 eV). We report similar conclusions for bilayer heterostructures in which the composition of the two layers is different. Interlayer registry also has a pronounced effect on piezoelectric properties as the piezoelectric coefficients of the two layers either nearly cancel each other or add up to yield enhanced values for the associated TMDC bilayer heterostructures. These results may serve as a guide for enhancing electronic and piezoelectric properties by stacking TMDC layers.