We describe a mechanism for pronounced biochemical oscillations, relevant to microscopic systems, such as the intracellular environment. This mechanism operates for reaction schemes which, when modeled using deterministic rate equations, fail to exhibit oscillations for any values of rate constants. The mechanism relies on amplification of the underlying stochasticity of reaction kinetics within a narrow window of frequencies. This amplification means that fluctuations have a dominant effect, even though the number of molecules in the system is relatively large. The mechanism is quantitatively studied within simple models of self-regulatory gene expression, and glycolytic oscillations.