The Korea Astronomy and Space Science Institute and the Department of Astronomy at the University of Texas at Austin are developing a near infrared wide-band high resolution spectrograph, immersion grating infrared spectrometer (IGRINS). The compact white-pupil design of the instrument optics uses seven cryogenic mirrors, including three aspherical off-axis collimators and four flat fold mirrors. In this study, we introduce the optomechanical mount designs of three off-axis collimating mirrors and one flat slit-viewer fold mirror. Two of the off-axis collimators are serving as H and K-band pupil transfer mirrors, and are designed as system alignment compensators in combination with the H2RG focal plane array detectors in each channel. For this reason, the mount designs include tip-tilt and parallel translation adjustment mechanisms to properly perform the precision alignment function. This means that the off-axis mirrors’ optomechanical mount designs are among the most sensitive tasks in all IGRINS system hardware. The other flat fold mirror is designed within its very limitedly allowed work space. This slit-viewer fold mirror is mounted with its own version of the six-point kinematic optics mount. The design work consists of a computer-aided 3D modeling and finite element analysis (FEA) technique to optimize the structural stability and the thermal behavior of the mount models. From the structural and thermal FEA studies, we conclude that the four IGRINS mirror mounts are well designed to meet all optical stability tolerances and system thermal requirements.