Tunable MEMS resonators having adjustable resonance frequency and capable of handling large signals are described. In one exemplary design, a tunable MEMS resonator includes (i) a first part having a cavity and a post and (ii) a second part mated to the first part and including a movable layer located under the post. Each part may be covered with a metal layer on the surface facing the other part. The movable plate may be mechanically moved by a DC voltage to vary the resonance frequency of the MEMS resonator. The cavity may have a rectangular or circular shape and may be empty or filled with a dielectric material. The post may be positioned in the middle of the cavity. The movable plate may be attached to the second part (i) via an anchor and operated as a cantilever or (ii) via two anchors and operated as a bridge.