We construct an implicit derivative matching (IDM) technique for restoring the accuracy of the Yee scheme for Maxwell's equations in dispersive media with material interfaces in one dimension. We consider media exhibiting orientational polarization, which are represented using a Debye dispersive model, examples of which are water and living tissue. The problems considered here have applications to noninvasive interrogation of complex materials, and microwave imaging of biological media, among others. The IDM technique employs fictitious points to locally modify the stencil of the Yee scheme to maintain the second order accuracy that the scheme exhibits in homogeneous media. Using numerical simulations we test the convergence of the IDM-Yee scheme and demonstrate its second order accuracy. We also discuss extensions of the IDM for modifying higher order staggered finite difference schemes for Maxwell's equations in heterogeneous dispersive media.