The correspondence rules of Miller, determining semiclassical approximations for general quantum amplitudes , are formulated in a way that is analogous to the initial value representation (IVR) treatments of the propagator. The semiclassical formulas obtained in this way do not require numerical searches, are free from caustic singularities, and are often uniform approximations. However, to develop treatments that retain the power and generality of Miller's rules, it is necessary to overcome boundary condition difficulties that arise when the x's are not Cartesian coordinates. Such problems can be solved by replacing Gaussian factors, analogous to those appearing in existing IVR treatments of the propagator, with more general non-Gaussian functions. It is suggested that factors of this kind can be obtained from certain IVR representations for wave functions that are exact for particular model systems. Examples of such factors are presented, and the resulting theory is illustrated. In one application, an IVR expression for the elastic scattering differential cross section that is uniform for all angles is developed and tested.