Heart valves are characterized to be highly flexible yet tough, and exhibit complex deformation characteristics such as nonlinearity, anisotropy, and viscoelasticity, which are, at best, only partially recapitulated in scaffolds for heart valve tissue engineering (HVTE). These biomechanical features are dictated by the structural properties and microarchitecture of the major tissue constituents, in particular collagen fibers. In this study, the unique capabilities of melt electrowriting (MEW) are exploited to create functional scaffolds with highly controlled fibrous microarchitectures mimicking the wavy nature of the collagen fibers and their load-dependent recruitment. Scaffolds with precisely-defined serpentine architectures reproduce the J-shaped strain stiffening, anisotropic and viscoelastic behavior of native heart valve leaflets, as demonstrated by quasistatic and dynamic mechanical characterization. They also support the growth of human vascular smooth muscle cells seeded both directly or encapsulated in fibrin, and promote the deposition of valvular extracellular matrix components. Finally, proof-of-principle MEW trileaflet valves display excellent acute hydrodynamic performance under aortic physiological conditions in a custom-made flow loop. The convergence of MEW and a biomimetic design approach enables a new paradigm for the manufacturing of scaffolds with highly controlled microarchitectures, biocompatibility, and stringent nonlinear and anisotropic mechanical properties required for HVTE.