  We construct classes in the motivic cohomology of certain 1-parameter
families of Calabi-Yau hypersurfaces in toric Fano n-folds, with applications
to local mirror symmetry (growth of genus 0 instanton numbers) and
inhomogeneous Picard-Fuchs equations. In the case where the family is
classically modular the classes are related to Belinson's Eisenstein symbol;
the Abel-Jacobi map (or rational regulator) is computed in this paper for both
kinds of cycles. For the "modular toric" families where the cycles essentially
coincide, we obtain a motivic (and computationally effective) explanation of a
phenomenon observed by Villegas, Stienstra, and Bertin.
