  We obtain an asymptotic formula for the mean value of L-functions associated
to cubic characters over F_q[t]. We solve this problem in the non-Kummer
setting when q=2 (mod 3) and in the Kummer case when q=1 (mod 3). The proofs
rely on obtaining precise asymptotics for averages of cubic Gauss sums over
function fields, which can be studied using the theory of metaplectic
Eisenstein series. In the non-Kummer setting we display some explicit
cancellation between the main term and the dual term coming from the
approximate functional equation of the L-functions.
