Thin-film silicon nitride (SiN) membranes mechanical resonators have been widely used for many fundamental opto-mechanical studies and sensing technologies due to their extremely low mechanical dissipation (high mechanical Q-factor). In this work, we experimentally demonstrate an opto-mechanical approach to perform thermal radiation sensing, using a SiN membrane resonator. An important aspect of this work is to develop a closed-form analytical heat transfer model for assessing the thermal coupling conditionbetween free-standing membranes and their environment. We also derive analytical expressions for other important intrinsic thermal quantities of the membrane, such as thethermal conductance, the heat capacity and the thermal time constant. Experimental results show good agreement with our theoretical prediction. Of central importance, we show that membranes of realistic dimensions can be coupled to their environment more strongly via radiation than by solid-state conduction. For example, membranes with 100nm thickness (frequently encountered size) are predicted to be radiation dominated when their side length exceeds 6 mm. Having radiation dominated thermal coupling is a key ingredient for reaching the fundamental detectivity limit of thermal detectors. Hence, our work proves that SiN membranes are attractive candidates for reaching the fundamental limit. We also experimentally exhibit the high temperature responsivity of the SiN membranes resonance, in which we shift a 88.7 KHz resonance by over 1 KHz when temperature increment on the membrane is approximately 2 K.