A graph is $s$-regular if its automorphism group acts regularly on the set of its $s$-arcs. In this paper, the cubic $s$-regular graphs of order 12p, 36p, 44p, 52p, 66p, 68p and 76p are classified for each $s{geq}1$ and each prime $p$. The number of cubic $s$-regular graphs of order 12p, 36p, 44p, 52p, 66p, 68p and 76p is 4, 3, 7, 8, 1, 4 and 1, respectively. As a partial result, we determine all cubic $s$-regular graphs of order 70p except for $p$ = 31, 41.