I present a recently developed Monte Carlo approach for simulating large numbers of strongly interacting fermions on the lattice. The method is applied to a system of two-component fermions at unitarity: the limit of infinite two-particle s-wave scattering length and vanishing interaction range. This nonrelativistic conformal system may be realized experimentally using trapped atoms tuned to a Feshbach resonance, and also provides a natural starting point for an effective field theory description of nuclear physics. Using the new approach, along with an unconventional method for extracting energies from multi-fermion correlation function distributions, we have performed studies of up to 66 fermions confined to a finite box, and 70 fermions confined to a harmonic trap. We compare few-body results for the energy of those systems with previously published benchmarks, as well as perform two independent many-body calculations of the Bertsch parameter, defined as the ground state energy of the unitary Fermi gas in units of the free gas energy at the same density. I will present numerical results from our studies, along with some potential future applications of our lattice technique.