Golf club heads include: (a) a body having a ball-striking face and a rear face, the body defining a cavity; (b) a bridge member extending across the cavity, wherein the bridge member is provided to affect, at least in part, a position of a center of gravity of the club head; and (c) a feel altering element provided between the bridge member and the rear face. The feel altering element may affect a wide variety of parameters that relate to the “feel” of the golf club head when it strikes a golf ball, such as the sound produced by the club head during the strike, the vibrational response of the club head during the strike, the swing characteristics of the club head (e.g., weight, center of gravity position, etc.), and the like.