In this paper, we study the rotation sensitivity of a gyroscope made of a two-dimensional array of coupled resonators consisting of N columns of one-dimensional coupled resonant optical waveguides (CROWs) connected by two bus waveguides, each CROW consisting of M identical ring resonators. We show that the maximum rotation sensitivity of this structure is a strong function of the parity of the number of rows M. For an odd number of rows, and when the number of columns is small, the maximum sensitivity is high, and it is slightly lower than the maximum sensitivity of a single-ring resonator with two input/output waveguides (the case M = N = 1), which is a resonant waveguide optical gyroscope (RWOG). For an even M and small N, the maximum sensitivity is much lower than that of the RWOG. Increasing the number columns N increases the sensitivity of an even-row 2D CROW sublinearly, as N0.39, up to 30 columns. In comparison, the maximum sensitivity of an RWOG of equal area increases faster, as √N. The sensitivity of the 2D CROW therefore always lags behind that of the RWOG. For a 2×2 CROW, if the spacing between the columns L is increased sufficiently the maximum sensitivity increases linearly with L due to the presence of a composite Mach- Zehnder interferometer in the structure. However, for equal footprints this sensitivity is also not larger than that of a single-ring resonator. Regardless of the number of rows and columns and the spacing, for the same footprint and propagation loss, a 2D CROW gyroscope is not more sensitive than an RWOG.