A near infrared wide-band high resolution spectrograph, immersion grating infrared spectrometer (IGRINS) has been jointly developed by the Korea Astronomy and Space Science Institute and the University of Texas at Austin. The compact white-pupil design of the instrument optics includes five cryogenic flat mirrors including a slit mirror, an input fold mirror, a dichroic mirror, and H&K camera fold mirrors. In this study, we introduce the optomechanical mount designs of the five cryogenic mirrors. In order to meet the structural stability and thermal requirements of the mount models, we conducted the design work with the aid of 3-dimensional computer modeling and the finite element analysis (FEA) method. We also present the actual fabricated parts and assemblies of the mounts and mirrors as well as their CAD models.