It is shown that electron cyclotron emission can be used for detailed diagnosis of mildly relativistic electron distribution functions, provided the plasma can be viewed along a line of constant magnetic field. Calculations of the emissivity are performed, both for tenuous and finite-background-density plasmas whose results allow observations of emission to be deconvolved in terms of the density and anisotropy of the distribution as a function of total electron energy. For prolate distributions different harmonics have to be measured, while for oblate distributions different polarizations have to be measured. Absorption can also be treated using these results. Some of the issues concerning experimental application are discussed.