Chains of magnetic atoms, placed on the surface of s-wave superconductors, have been established as a laboratory for the study of Majorana bound states. In such systems, the breaking of time reversal due to magnetic moments gives rise to the formation of in-gap states, which hybridize to form one-dimensional topological superconductors. However, in unconventional superconductors even nonmagnetic impurities induce in-gap states since scattering of Cooper pairs change their momentum but not their phase. Here we propose a path for creating topological superconductivity, which is based on an unconventional superconductor with a chain of nonmagnetic adatoms on its surface. The topological phase can be reached by tuning the magnitude and direction of a Zeeman field, such that Majorana zero modes at its boundary can be generated, moved, and fused. To demonstrate the feasibility of this platform, we develop a general mapping of films with adatom chains to one-dimensional lattice Hamiltonians. This allows us to study unconventional superconductors such as Sr2RuO4 exhibiting multiple bands and an anisotropic order parameter.