We address the properties of (1+1)-dimensional periodic waves in conservative saturable cubic nonlinear media and discover that cnoidal- and snoidal-type waves are completely stable within a broad range of parameters. The existence of stability bands is in sharp contrast with the previously known properties of periodic waves in self-focusing Kerr nonlinear media. We also found that in self-defocusing media instability bands occur, again in contrast to the case of Kerr media.