This paper continues our studies of quantum field theories on a lattice. We develop techniques for computing the low-lying spectrum of a lattice Hamiltonian using a variational approach, without recourse either to weak- or strong-coupling expansions. Our variational methods, which are relatively simple and straightforward, are applied to the Ising model in a transverse magnetic field as well as to a free spinless field theory. We demonstrate their accuracy in the vicinity of a phase transition for the Ising model by comparing with known exact solutions.