  Motivated by the problem of polaron effect due to phonon-modulated hopping,
we formulate a generic Monte Carlo technique for solving it in the coordinate
representation for both the particle and atomic displacements. The method
applies to a broad class of models; the only condition is that the hopping
amplitude be sign-positive. A dramatic simplification of the scheme, with the
corresponding efficiency gain, takes place in models with dispersionless
phonons. Our study sheds important light on the nature and universality of the
most striking qualitative and quantitative effects demonstrated by the
"standard" (Peierls/Su-Schrieffer-Heeger) model based on the linearized
displacement-modulated hopping.
