Rigid closed-cell polyurethane (PUR) foams are used in diverse applications ranging from thermal insulation in buildings and appliances and cryogenic insulation in liquefied gas tankers and airspace vehicles to impact-mitigating components in the automotive industry. To alleviate environmental concerns and enhance the sustainability, polyols derived from renewable or recycled resources should be used in foam production. In this study, the mechanical response of PUR foams and the respective monolithic polymers with polyols derived by different processes from rapeseed oil, tall oil, or PET manufacturing side stream products is considered. The effect of organoclay fillers on the stiffness and strength of the foams is evaluated. The mechanical properties of foams are found to compare favorably to those containing polyols of a solely petrochemical origin. The applicability of models of the mechanical properties of the foams is evaluated for predicting the Young’s modulus, Poisson’s ratio, and strength of the PUR foams as functions of porosity varying in the high to medium range. The stiffness and strength of neat and filled foams is found to rather closely correlate with the respective characteristics of the monolithic neat and filled PUR.