The transmission of surface plasmon polaritons through a one-dimensional periodic structure is considered theoretically by using the transfer matrix approach. The periodic structure is assumed to have alternate left-handed metamaterial and dielectric layers. Both transverse electric and transverse magnetic modes of surface plasmon polaritons exist in this structure. It is found that, for nonlinear wave propagation, tunneling structures are formed to transform nontransmitting frequencies into transmitting frequencies and hence transmission bistability is observed. It is further observed that the structure shows sensitivity with respect to the polarization of the electromagnetic field for this phenomenon.