We show that in collisions with neutron-rich heavy ions at energies around the production threshold K0 and K+ yields probe the isospin-dependent part of the nuclear equation of state at high baryon densities. In particular, we suggest the K0/K+ ratio as a promising observable. Results obtained in a covariant relativistic transport approach are presented for Au+Au collisions at 0.8-1.8A GeV. The focus is put on the equation of state influence which goes beyond the collision-cascade picture. The isovector part of the in-medium interaction affects the kaon multiplicities via two mechanisms: (i) a symmetry potential effect, i.e., a larger neutron repulsion in n-rich systems, and (ii) a threshold effect, due to the change in the self-energies of the particles involved in inelastic processes. Genuine relativistic contributions are revealed that could allow one to directly "measure" the Lorentz structure of the effective isovector interaction.