  We obtain explicit formulas for enumerating $3$-regular one-face maps on
orientable and non-orientable surfaces of a given genus $g$ up to all
symmetries. We use recent analytical results obtained by Bernardi and Chapuy
for counting rooted precubic maps on non-orientable surfaces together with more
widely known formulas for counting precubic maps on orientable surfaces. To
take into account all symmetries we use a result of Krasko and Omelchenko that
allows to reduce this problem to the problem of counting rooted quotient maps
on orbifolds.
