Construction of synthetic biochemical circuits from simple components provides a useful approach to establish the minimal determinants required for complex logical functions. As stripped-down analogues of cellular regulatory networks, we engineered artificial transcriptional oscillators consisting of synthetic DNA switches regulated by RNA signals, and two enzymes, bacteriophage T7 RNA polymerase and Escherichia coli ribonuclease H. The switches support sharp and adjustable thresholds using a competitive hybridization mechanism, analogous to a biological threshold mechanism, "inhibitor ultrasensitivity", thus allowing arbitrary analog or digital circuits to be created in principle.