  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
non-magnetic impurities induce in-gap states since scattering of Cooper pairs
changes their momentum but not their phase. Here, we propose a realistic path
for creating topological superconductivity, which is based on an unconventional
superconductor with a chain of non-magnetic 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
Sr$_2$RuO$_4$ exhibiting multiple bands and an anisotropic order parameter.
