We study a classical K-type three-level system based on three high-Q micromechanical beam resonators embedded in a gradient electric field. By modulating the strength of the field at the difference frequency between adjacent beam modes, we realize strong dynamic two-mode coupling, via the dielectric force. Driving adjacent pairs simultaneously, we observe the formation of a purely mechanical "dark" state and an all-phononic analog of coherent population trapping-signatures of strong three-mode coupling. The K-type micromechanical system is a natural extension of previously demonstrated "two-level" micromechanical systems and adds to the toolbox for engineering of all-phononic micromechanical circuits and arrays. (C) 2016 AIP Publishing LLC.