  We show for several two-dimensional lattices that the nearest neighbor
valence bond states are linearly independent. To do so, we utilize and
generalize a method that was recently introduced and applied to the kagome
lattice by one of the authors. This method relies on the choice of an
appropriate cell for the respective lattice, for which a certain local linear
independence property can be demonstrated. Whenever this is achieved, linear
independence follows for arbitrarily large lattices that can be covered by such
cells, for {\em both} open and periodic boundary conditions. We report that
this method is applicable to the kagome, honeycomb, square, squagome, two types
of pentagonal, square-octagon, the star lattice, two types of archimedean
lattices, three types of "martini" lattices, and to fullerene-type lattices,
e.g., the well known "Buckyball". Applications of the linear independence
property, such as the derivation of effective quantum dimer models, or the
constructions of new solvable spin-1/2 models, are discussed.
