Oscillations within the cell regulate the timing of many important life cycles. However, in this noisy environment, oscillations can be highly inaccurate owing to phase fluctuations. It remains poorly understood how biochemical circuits suppress these phase fluctuations and what is the incurred thermodynamic cost. Here, we study three different types of biochemical oscillation, representing three basic oscillation motifs shared by all known oscillatory systems. In all the systems studied, we find that the phase diffusion constant depends on the free-energy dissipation per period, following the same inverse relation parameterized by system-specific constants. This relationship and its range of validity are shown analytically in a model of noisy oscillation. Microscopically, we find that the oscillation is driven by multiple irreversible cycles that hydrolyse fuel molecules such as ATP; the number of phase coherent periods is proportional to the free energy consumed per period. Experimental evidence in support of this general relationship and testable predictions are also presented.