Non-Hermitian systems have attracted much attention during the past few years, both theoretically and experimentally. The existence of non-Hermiticity can induce multiple exotic phenomena that cannot be observed in Hermitian systems. In this work, we introduce a new non-Hermitian system called the non-Hermitian mosaic dimerized lattice. Unlike the regular nonreciprocal lattices where asymmetric hoppings are imposed on every hopping term, here in the mosaic dimerized lattices the staggered asymmetric hoppings are only added to the nearest-neighboring hopping terms with equally spaced sites. By investigating the energy spectra, the non-Hermitian skin effect (NHSE), and the topological phases in such lattice models, we find that the period of the mosaic asymmetric hopping can influence the system’s properties significantly. For a system with real system parameters, we find that as the strength of asymmetric hopping increases, the energy spectra of the system under open boundary conditions will undergo a real-imaginary or real-complex transition. As to the NHSE, we find that when the period is odd, there appears no NHSE in the system and the spectra under open boundary conditions (OBCs) and periodic boundary conditions (PBCs) are the same (except for the topological edge modes under OBCs). If the period of the mosaic asymmetric hopping is even, the NHSE will emerge and the spectra under different boundary conditions exhibit distinctive structures. The PBC spectra form loop structures, indicating the existence of point gaps that are absent in the spectra under OBCs. The point gap in the PBC spectrum is shown to be the topological origin of the NHSE under OBCs, which also explains the NHSE in our mosaic dimerized lattices. To distinguish whether the bulk states of the system under OBCs are shifted to the left or right end of the one-dimensional lattice due to the NHSE, we define a new variable called the directional inverse participation ratio (dIPR). The positive dIPR indicates that the state is localized at the right end while the negative dIPR corresponds to the states localized at the left end of the one-dimensional lattice. We further study the topological zero-energy edge modes and characterize them by calculating the Berry phases based on the generalized Bloch Hamiltonian method. In addition, we also find that the topological edge modes with nonzero but constant energy can exist in the system. Our work provides a new non-Hermitian lattice model and unveils the exotic effect of mosaic asymmetric hopping on the properties of non-Hermitian systems.