In this first paper of a series dedicated to the microphysics of unmagnetized, relativistic collisionless pair shocks, we discuss the physics of the Weibel-type transverse current filamentation instability that develops in the shock precursor, through the interaction of an ultrarelativistic suprathermal particle beam with the background plasma. We introduce in particular the notion of the "Weibel frame," or scattering center frame, in which the microturbulence is of mostly magnetic nature. We calculate the properties of this frame, using first a kinetic formulation of the linear phase of the instability, relying on Maxwell-Jüttner distribution functions, then using a quasistatic model of the nonlinear stage of the instability. Both methods show that (i) the Weibel frame moves at subrelativistic velocities relative to the background plasma, therefore at relativistic velocities relative to the shock front; (ii) the velocity of the Weibel frame relative to the background plasma scales with ξ_{b}, i.e., the pressure of the suprathermal particle beam in units of the momentum flux density incoming into the shock; and (iii) the Weibel frame moves slightly less fast than the background plasma relative to the shock front. Our theoretical results are found to be in satisfactory agreement with the measurements carried out in dedicated large-scale 2D3V particle-in-cell simulations.