We present an algebraic multigrid method for the solution of linear (block-)systems coming from a discretization of a system of partial differential equations. It generalizes the classical AMG approach for scalar problems to systems of PDEs in a natural blockwise fashion. We show that the block interpolation, described in this paper, reproduces the rigid body modes, i.e., the kernel elements of the linear elasticity operator. It is well known from geometric multigrid methods that this reproduction of the kernel elements is an essential property to obtain optimal convergence rates which are independent of the problem size. Finally the presentation of numerical experiments confirms that the method is robust and stable with fast convergence for a variety of discretized elasticity problems.