We report the experimental observation of traveling concentration waves and spirals in a chemical reaction network built from the bottom up. The mechanism of the network is an oscillator of the predator-prey type, and this is the first time that predator-prey waves have been observed in the laboratory. The molecular encoding of the nonequilibrium behavior relies on small DNA oligonucleotides that enforce the network connectivity and three purified enzymes that control the reactivity. Wave velocities in the range 80-400 μm min(-1) were measured. A reaction-diffusion model in quantitative agreement with the experiments is proposed. Three fundamental parameters are easy to tune in nucleic acid reaction networks: the topology of the network, the rate constants of the individual reactions, and the diffusion coefficients of the individual species. For this reason, we expect such networks to bring unprecedented opportunities for assaying the principles of spatiotemporal order formation in chemistry.