The technological challenges required to discover earthlike planets orbiting sunlike stars have nearly all been solved. A new generation of instruments, such as EXPRES, ESPRESSO, and NEID, will be able to achieve the required radial velocity measurement precision of 10 cm/s. Yet searches for earthlike, habitable planets are plagued by false positives: discoveries of “planets” that later turn out to be parasitic signals from stellar activity, windowing, or data analysis methods. Here we extend the multitaper time-series analysis technique (Thomson 1982, Chave 2019) to unevenly sampled data. By pre-multipling radial velocity measurements with optimal data tapers, we constrain both the bias and variance of the periodogram estimator. Our multitaper periodogram has improved statistical properties and lower false-positive rates than the generalized Lomb-Scargle periodogram.