We study absorption and scattering by irregularly shaped Gaussian random particles in the Rayleigh-ellipsoid approximation. For a given sample shape, we determine the best-fitting ellipsoid as the equal-volume ellipsoid with the largest volume overlapping the sample shape. We present an efficient method for calculating such ellipsoids for Gaussian particles and characterize the goodness of the approximation with the complementary volume. We study the scattering properties of Gaussian particles much smaller than the wavelength with different complex refractive indices, comparing the Rayleigh-ellipsoid approximation to the Rayleigh-volume, discrete-dipole, and second-order perturbation approximations, and to the computations using the variational volume integral equation method. Our new method can prove valuable in microwave remote sensing of terrestrial ice clouds: crystalline structures are often elongated with dimensions in the Rayleigh domain for typical radar frequencies.