We describe an electromagnetic (EM) composite material that possesses subwavelength tunneling characteristics. The sample is constructed as a multilayered "sandwich",i.e., ABA structure. The core plate is a metallic mesh with holes subwavelength in size compared to the incident EM wavelength, while the two outer layers are fabricated with units of periodically arranged positive dielectric permittivity or negative magnetic permeability. Up to 100% transmission of normally incident EM waves has been demonstrated in the microwave regime through such "sandwiches" at certain frequencies, forming so called "passbands". The phenomenon of subwavelength tunneling is caused by the electromagnetic enhancement at and between the interfaces of the different layers, induced by local resonances of the two outer plates.