For a smooth and proper variety Y over a finite field k the reciprocity map rho(Y) : CH0(Y) -> pi(ab)(1) (Y) is injective with dense image. For a proper simple normal crossing variety this is no longer true in general. In this paper we give a description of the kernel and cokernel of the reciprocity map in terms of homology groups of a complex filled with descent data using an algebraic Seifert-van Kampen theorem. Furthermore, we give a new criterion for the injectivity of the reciprocity map for proper simple normal crossing varieties over finite fields.