Let G be a countable group acting by measure-preserving transformations on a probability space (X,μ). To every finite subset of G there is an associated averaging operator on L^p(X,μ). Ergodic theorems describe the pointwise and mean limits of sequences of such operators. There is a very complete result due to Lindenstrauss in the special case in which G is amenable and the sequence of subsets satisfies the (tempered) Følner property. We extend this result to an arbitrary group G which has a `nice' amenable action and obtain specialized results when G is Gromov hyperbolic.