An extension of the essentially nonoscillatory formulation via extrapolation on unstructured meshes to systems of conservation laws is presented. Higher-order accuracy is obtained using a piecewise polynomial reconstruction of the solution from its cell averages. To avoid spurious oscillations near discontinuities and maintain accuracy in smooth regions, the reconstruction procedure selects an adaptive stencil for each cell. The reconstruction via extrapolation allows the selection in one step of all of the cells in the required stencil for a given order. This leads to important savings in computer time and makes the essentially nonoscillatory schemes on unstructured meshes suitable for practical applications. The problems encountered when using triangular meshes, as well as possible ways to overcome them, are discussed. Numerical results and comparison of the efficiency against other schemes are also presented