We present a micro-electro-mechanical system-based experimental technique to measure thermal conductivity of freestanding ultra-thin films of amorphous silicon nitride (Si3N4) as a function of mechanical strain. Using a combination of infrared thermal micrography and multi-physics simulation, we measured thermal conductivity of 50 nm thick silicon nitride films to observe it decrease from 2.7 W (m K)(-1) at zero strain to 0.34 W (m K)(-1) at about 2.4% tensile strain. We propose that such strong strain-thermal conductivity coupling is due to strain effects on fraction-phonon interaction that decreases the dominant hopping mode conduction in the amorphous silicon nitride specimens.