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We measure the frequency spectra of random spin fluctuations, or “spin noise,” in ensembles of (In, Ga) As/GaAs quantum dots (QDs) at low temperatures. We employ a spin noise spectrometer based on a sensitive optical Faraday rotation magnetometer that is coupled to a digitizer and field-programmable gate array, to measure and average noise spectra from 0–1 GHz continuously in real time with subnanoradian/Hz sensitivity. Both electron and hole spin fluctuations generate distinct noise peaks, whose shift and broadening with magnetic field directly reveal their g factors and dephasing rates within the ensemble. A large, energy-dependent anisotropy of the in-plane hole g factor is clearly exposed, reflecting systematic variations in the average QD confinement potential.