Electron-phonon coupling is studied for acoustically non-uniform layered nanostructures consisting of a metal film deposited on an insulating membrane. Electrons are heated by an external source to a temperature higher than the temperature of the crystal lattice ( Te > Tp ) and the resulting heat flux transferred from electrons to phonons is considered for a low temperature regime. The system is representative of the central element in the electronic microrefrigerators designed for the on-chip cooling of ultrasensitive detectors far down to the sub-Kelvin region [1]. For the total thickness below 100-200 nm the phonon subsytem is dominated by long wavelength vibrations and can be treated in terms of an effectively quasi-two-dimensional elastic medium with a dramatic enhancement of the heat transfer over the bulk material [2,3]. By assuming the deformation potential mechanism of coupling, an explicit expression for the electron-phonon heat flux is derived analytically by taking into account the acoustic non-uniformity of the device.   The obtained results explain the behavior observed in some experiments including the case of Cu film supported by a SiN insulating membrane. It is shown that the main contribution to the heat flux is due to the electron coupling with Lamb’s dilatational and flexural acoustic modes in a roughly comparable amount. The role of the flexural modes increases as the thickness of the film decreases.