  We develop a technique for calculating three-dimensional classical partition
functions using projected entangled-pair states (PEPS). Our method is based on
variational PEPS optimization algorithms for two-dimensional quantum spin
systems, and allows us to compute free energies directly in the thermodynamic
limit. The main focus of this work is classical frustration in
three-dimensional many-body systems leading to an extensive ground-state
degeneracy. We provide high-accuracy results for the residual entropy of the
dimer model on the cubic lattice, water-ice $I_h$ and water-ice $I_c$. In
addition, we show that these systems are in a Coulomb phase by computing the
dipolar form of the correlation functions. As a further benchmark of our
methods, we calculate the critical temperature and exponents of the Ising model
on the cubic lattice.
