  We introduce a new method for computing massless Feynman integrals
analytically in parametric form. An analysis of the method yields a criterion
for a primitive Feynman graph $G$ to evaluate to multiple zeta values. The
criterion depends only on the topology of $G$, and can be checked
algorithmically. As a corollary, we reprove the result, due to Bierenbaum and
Weinzierl, that the massless 2-loop 2-point function is expressible in terms of
multiple zeta values, and generalize this to the 3, 4, and 5-loop cases. We
find that the coefficients in the Taylor expansion of planar graphs in this
range evaluate to multiple zeta values, but the non-planar graphs with crossing
number 1 may evaluate to multiple sums with $6^\mathrm{th}$ roots of unity. Our
method fails for the five loop graphs with crossing number 2 obtained by
breaking open the bipartite graph $K_{3,4}$ at one edge.
