We report measurements of the equilibrium dynamics of fluctuating concentration in a polymer bicontinuous microemulstion. The microemulsion sample consists of a symmetric blend of polystyrene (PS) and polyisoprene (PI), compatibilized by a corresponding symmetrick PS-PI diblock copolymer. Molecular weights and compositions are adjusted to access an equilibrium bicontinuous microemulsion phase. Although the individual homopolymer constituents are nearly Newtonian owing to their low molecular weight, the microemulsion exhibits pronounced viscoelasticity in small-amplitude oscillatory testing, owing to the interconnected morphology at length scales of 10s of nanometers. Exploiting the high coherence of the Advance Photon Source, x-ray photon correlation spectroscopy (XPCS) was used to study the dynamics of concentration fluctuations in this sample. At the temperatures studied, the autocorrelation function of the structure factor exhibits stretched-exponential relaxation behavior, with a relaxation time that depends strongly on scattering vector and temperature within the bicontinuous microemulsion phase. These results are compared to a time-dependent extension of the Teubner-Strey free energy expansion that is frequently used to describe the static structure factor of bicontinuous microemulsion samples. Further, we attempt to use this theory to connect structural dynamics measured by XPCS to the linear viscoelastic response of this sample.