The domain-growth properties of the two-dimensional single spin-flip kinetic Ising model with a Metropolis-type transition function and checkerboard updating are investigated numerically with quenches from the high-temperature phase to temperatures in the ordered phase and to T(c). Measuring the structure factor S(0, t) is-proportional-to t2x and the autocorrelation function A(t) is-proportional-to t(-lambdaz) as a function of time, the effect of deterministic dynamics at T = 0 with x = 1.0 is followed throughout the ordered phase. At T < T(c), T almost-equal-to T(c), crossover from x = 1.0 to x = 0.5 is shown to occur with increasing time and even at T = T(c) the scaling exponents are still discernibly dependent on time. Lambda, however, is found to be unaffected in the whole temperature range.