We use scanning fluorescence microscopy and electron beam lithography to probe the mechanism of fluorescence enhancement by periodic arrays of silver nanostructures, determining the optimum size and spacing of both Ag nanowires and Ag nanocolumns for incident light of different wavelengths and polarizations. Finite difference time domain (FDTD) calculations show a systematic variation with spatial period and incident polarization of the local electric field above the surface of the arrays which correlate well with that of the measured fluorescence enhancement, but a lack of a simple proportionality indicates that the dependence of the radiative and nonradiative decay rates on array geometry must be included in models for this effect. The dependence of the enhancement on spatial period and polarization indicates the importance of surface plasmon standing waves in this effect.