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Based on the well-established theory of discrete conjugate nets in discretedifferential geometry, we propose and examine discrete analogues of importantobjects and notions in the theory of semi-Hamiltonian systems of hydrodynamictype. In particular, we present discrete counterparts of (generalised)hodograph equations, hyperelliptic integrals and associated cycles,characteristic speeds of Whitham type and (implicitly) the correspondingWhitham equations. By construction, the intimate relationship with integrablesystem theory is maintained in the discrete setting.

Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations