We develop a new sampling strategy that uses the hit-and-run algorithm within level sets of a target density. Our method can be applied to any quasi-concave density, which covers a broad class of models. Standard sampling methods often perform poorly on densities that are high-dimensional or multi-modal. Our level set sampler performs well in high-dimensional settings, which we illustrate on a spike-and-slab mixture model. We also extend our method to exponentially-tilted quasi-concave densities, which arise in Bayesian models consisting of a log-concave likelihood and quasiconcave prior density. We illustrate our exponentially-tilted level-set sampler on a Cauchy-normal model where our sampler is better able to handle a high-dimensional and multi-modal posterior distribution compared to Gibbs sampling and Hamiltonian Monte Carlo.