Models of the chemistry of dense interstellar clouds are presented in which both gas-phase and grain-surface chemistry occur. The dust grain and gas temperatures are fixed at 10 K, and the gas density n = n(H) + 2n(H-2) remains approximately at 2 x 10(4) cm-3 in these models, which are designed primarily to represent the chemistry occurring in dark clouds. We utilize previous ideas on what constitutes the most likely reactions to occur on large classical grains. Grain reactions that produce molecules as complex as those in our previous models of gas-phase chemistry are included to help elucidate the role of grains in the synthesis of organic molecules. The only desorption process allowed is thermal evaporation, so that heavy neutral species remain on the grain surfaces. The time-dependent gas-grain chemistry is followed by solving coupled differential equations. Three different sets of initial gas-phase abundances are assumed: steady state abundances, as calculated in our earlier gas-phase models; neutral atomic abundances; and the standard initial abundances utilized in pseudo-time-dependent gas-phase calculations, in which hydrogen is mainly molecular. Results as a function of time are presented for the different initial cloud compositions and compared both with previous, more restricted calculations of this type and with available observations. Among our conclusions is that a homogeneous gas-grain model for dark interstellar clouds can represent adequately observed abundances on grain surfaces (e.g., H2O, CH4) as well as in the gas at so-called early time. We also conclude that significant abundances of complex organic molecules can be synthesized on grain surfaces from gaseous precursors of varying complexity.