  A suspended system for measuring the thermal properties of membranes is
presented. The sensitive thermal measurement is based on the 3$\omega$ dynamic
method coupled to a V$\ddot{o}$lklein geometry. The device obtained using
micro-machining processes allows the measurement of the in-plane thermal
conductivity of a membrane with a sensitivity of less than 10nW/K
(+/-$5x10^{-3}$Wm$^{-1}K^{-1}$ at room temperature) and a very high resolution
($\Delta K/K =10^{-3}$). A transducer (heater/thermometer) centered on the
membrane is used to create an oscillation of the heat flux and to measure the
temperature oscillation at the third harmonic using a Wheatstone bridge set-up.
Power as low as 0.1nanoWatt has been measured at room temperature. The method
has been applied to measure thermal properties of low stress silicon nitride
and polycrystalline diamond membranes with thickness ranging from 100 nm to 400
nm. The thermal conductivity measured on the polycrystalline diamond membrane
support a significant grain size effect on the thermal transport.
