We present a theoretical investigation into the energy transport and transient wave propagation in the metamaterial tunneling structures consisting of ε-negative (ENG) and μ-negative (MNG) materials. It is proved that the conjugated matched ENG/MNG bilayer and the (zero index material doped) photonic crystal heterostructure can work as a sub-wavelength resonator at tunneling frequency. While the tunneling modes need a certain time to achieve to steady state and the characteristic time increase (nearly) exponentially with the thickness of the ENG/MNG bilayer, while the characteristic time of the (zero index material doped) photonic crystal heterostructure is lager than that of ENG/MNG bilayer because of multiple interface reflection. Under the steady state, the waves in the single negative material structures are not evanescent, but a hybrid of traveling wave and reactive standing wave, and the phase difference between the electric field and magnetic field varies with the position.