We define higher-dimensional linear rewriting systems, called linear polygraphs, for presentations of associative algebras, generalizing the notion of noncommutative Gröbner bases. They are constructed on the notion of category enriched in higher-dimensional vector spaces. Linear polygraphs allow more possibilities of termination orders than those associated to Gröbner bases. We introduce polygraphic resolutions of algebras giving a description obtained by rewriting of higher-dimensional syzygies for presentations of algebras. We show how to compute polygraphic resolutions starting from a convergent presentation, and how to relate these resolutions with the Koszul property of algebras.