We study an index calculus algorithm to solve the discrete logarithm problem (DLP) in degree 0 class groups of non-hyperelliptic curves of genus 3 over finite fields. We present a heuristic analysis of the algorithm which indicates that the DLP in degree 0 class groups of non-hyperelliptic curves of genus 3 can be solved in an expected time of Õ(q). This heuristic result relies on one heuristic assumption which is studied experimentally. We also present experimental data which show that a variant of the algorithm is faster than the Rho method even for small group sizes, and we address practical limitations of the algorithm.