We have measured the thermal conductivity of the geometrically frustrated quasi-one-dimensional spin system Cu3Mo2O9 in magnetic fields. A contribution of the thermal conductivity due to spins has been observed in the thermal conductivity along the spin chains. The thermal conductivity due to phonons, κphonon, has been found to decrease by the application of a magnetic field, which has been explained as being due to the reduction in the spin gap originating from the spin-singlet dimers. Moreover, it has been found that κphonon increases with increasing field in high fields above ∼7T at low temperatures. This suggests the existence of a novel field-induced spin state and is discussed in terms of the possible spin-chirality ordering in a frustrated Mott insulator.