A variety of light-emitting devices for general illumination utilizing solid state light sources (e.g., light emitting diodes) are disclosed. In general, the devices include a scattering element in combination with an extractor element. The scattering element, which may include elastic and/or inelastic scattering centers, is spaced apart from the light source element. Opposite sides of the scattering element have asymmetric optical interfaces, there being a larger refractive index mismatch at the interface facing the light emitting element than the interface between the scattering element and the extractor element. Such a structure favors forward scattering of light from the scattering element. In other words, the system favors scattering out of the scattering element into the extractor element over backscattering light towards the light source element. The extractor element, in turn, is sized and shaped to reduce reflection of light exiting the light-emitting device at the devices interface with the ambient environment.