We show that the eigenmodes for electromagnetic waves in an inhomogeneous dielectric medium can be obtained with an algorithm that scales linearly with the size of the system. The method employs discretization of the Maxwell equations in both the spatial and the time domain and the integration of the Maxwell equations in the time domain. The spectral intensity can then be obtained by a Fourier transform. We applied the method to a few problems of current interest, including the photonic band structure of a periodic dielectric structure, the effective dielectric constants of some three-dimensional and two-dimensional systems, and the defect states of a periodic dielectric structure with structural defects.