We present ASCA temperature profiles and, when possible, crude temperature maps for a sample of bright clusters with 0.04<z<0.09. Together with several previously published clusters, the sample includes A85, A119, A399, A401, A478, A644, A754, A780, A1650, A1651, A1795, A2029, A2065, A2142, A2256, A2319, A2597, A2657, A3112, A3266, A3376, A3391, A3395, A3558, A3571, A3667, A4059, Cygnus A, MKW3S, and Triangulum Australis. Nearly all clusters show a significant radial temperature decline. For a typical 7 keV cluster, the temperature decline between 1 and 6 X-ray core radii (0.15 and 0.9/h Mpc) can be approximately quantified by a polytropic index of 1.2-1.3. Assuming such a polytropic temperature profile, the gravitating mass within 1 and within 6 core radii is approximately 1.35 and 0.7 times the isothermal beta-model estimates, respectively. Most interestingly, we find that temperature profiles, excluding those for the most asymmetric clusters, appear remarkably similar when plotted against radius in units of the estimated virial radius. We compare the composite temperature profile to the published hydrodynamic simulations. The observed profiles appear steeper than those in most Lagrangian simulations (Evrard etal 1996; Eke etal 1997). The predictions for Omega=1 models are most discrepant, while models with low Omega are closer to our data. We note, however, that at least one Omega=1 Lagrangian simulation (Katz & White 1993) and the recent high-resolution Eulerian simulation (Bryan & Norman 1997) produced clusters with temperature profiles similar to or steeper than those observed. Our results thus provide a new constraint for adjusting numerical simulations and, potentially, discriminating among models of cluster formation. (ABRIDGED)