A new frequency domain iterative three-dimensional surface integral equation method (SIE) is proposed and tested in a number of electromagnetic computation problems. In contrast to the usual surface current SIE formulation, it is formulated by applying the scalar Green's theorem to the three Cartesian components of the magnetic field independently. The boundary conditions are implemented iteratively. Numerical results are given in three areas of electromagnetic computation: eddy current induction, EM wave scattering off dielectric targets, and the EM response of a conductor in a conductive host. Some of our numerical results involve two boundaries. Comparisons are made with previously published results where available, and close quantitative agreement is found. The numerical results show that the method can handle a very wide range in conductivity (up to 1015) and in dielectric constant (up to 150) without encountering numerical instability. Whether the method will always yield a convergent solution for different kinds of EM problems is yet to be studied, but for problems for which convergence can be achieved, we found the method has advantages in terms of computer operations and storage over existing techniques.