Two-dimensional electromagnetic crystals formed by rectangular lattices of thin ideally conducting cylinders periodically loaded by bulk reactive impedances are considered. An analytical theory of dispersion and reflection from this medium is presented. The consideration is based on the local field approach. The transcendental dispersion equation is obtained in the closed form and solved numerically. Different types of the loads such as inductive, capacitive, serial, and parallel LC circuits are considered. Typical dispersion curves and reflection coefficients are calculated and analyzed.