The recent advent of two-dimensional monolayer materials with tunable optoelectronic properties and high carrier mobility offers renewed opportunities for efficient, ultra-thin excitonic solar cells alternative to those based on conjugated polymer and small molecule donors. Using first-principles density functional theory and many-body calculations, we demonstrate that monolayers of hexagonal BN and graphene (CBN) combined with commonly used acceptors such as PCBM fullerene or semiconducting carbon nanotubes can provide excitonic solar cells with tunable absorber gap, donor-acceptor interface band alignment, and power conversion efficiency, as well as novel device architectures. For the case of CBN-PCBM devices, we predict the limit of power conversion efficiencies to be in the 10 - 20% range depending on the CBN monolayer structure. Our results demonstrate the possibility of using monolayer materials in tunable, efficient, polymer-free thin-film solar cells in which unexplored exciton and carrier transport regimes are at play.