An elastic metamaterial with a low-frequency passband is proposed by imitating a lattice system with linear on-site potential. It is shown that waves can only propagate in the tunable passband. Then, two kinds of elastic metamaterials with double passbands are designed. Great wave attenuation performance can be obtained at frequencies between the two passbands for locally resonant type metamaterials, and at both low and high frequencies for the diatomic type metamaterials. Finally, the strategy to design two-dimensional (2D) metamaterials is demonstrated. The present method can be used to design new types of small-size waveguides, filters, and other devices for elastic waves.