  Ising model is a famous toy model in condensed matter and statistical
physics. In this work we present a free-fermion formulation of the
two-dimensional classical Ising models on the honeycomb, triangular and
Kagom\'e lattices. Each Ising model is studied in the case of a zero field and
the case of an imaginary field i({\pi}/2)k_BT. We employ the decorated lattice
technique, star-triangle transformation and weak-graph expansion method to
exactly map each Ising model in both cases into an eight-vertex model on the
square lattice. The resulting vertex weights are shown to satisfy the
free-fermion condition. In the zero field case, each Ising model is an even
free-fermion model. In the case of the imaginary field, the Ising model on the
honeycomb lattice is an even free-fermion model while those on the triangular
and Kagom\'e lattices are odd free-fermion models. The exact solution of the
Kagom\'e lattice Ising model in the imaginary field i({\pi}/2)k_BT is obtained,
which has not been reported in previous works. The frustrated Ising models on
the triangular and Kagom\'e lattices in the imaginary field still exhibit a
non-zero residual entropy.
