Spin waves, the collective excitations of the magnetic order parameter, and magnons, the associated quasiparticles, are envisioned as possible data carriers in future wave-based computing architectures. On the road toward spin-wave computing, the development of a diodelike device capable of transmitting spin waves in only one direction, thus allowing controlled signal routing, is an essential step. Here we report on the design and experimental realization of a microstructured magnonic diode in a ferromagnetic bilayer system. Effective unidirectional propagation of spin waves is achieved by taking advantage of nonreciprocities produced by dynamic dipolar interactions in transversally magnetized media, which lack symmetry about their horizontal midplane. More specifically, dipolar-induced nonreciprocities are used to engineer the spin-wave dispersion relation of the bilayer system so that the group velocity is reduced to very low values for one direction of propagation and not for the other, thus producing unidirectional slow spin waves. Brillouin light scattering and propagating-spin-wave spectroscopy are used to demonstrate the diodelike behavior of the device, the composition of which is first optimized through micromagnetic simulations.