Normalizing flows are a popular approach for constructing probabilistic and generative models. However, maximum likelihood training of flows is challenging due to the need to calculate computationally expensive determinants of Jacobians. This paper takes steps towards addressing this challenge by introducing objectives and model architectures for determinant-free training of flows. Central to our framework is the energy objective, a multidimensional extension of proper scoring rules that admits efficient estimators based on random projections. The energy objective does not require calculating determinants and therefore supports general flow architectures that are not well-suited to maximum likelihood training. In particular, we introduce semi-autoregressive flows, an architecture that can be trained with the energy loss, and that interpolates between fully autoregressive and non-autoregressive models, capturing the benefits of both. We empirically demonstrate that energy flows achieve competitive generative modeling performance while maintaining fast generation and posterior inference.