A graph is

-regular if its automorphism group acts regularly on the set of its

-arcs. In this paper, the cubic

-regular graphs of order 12p, 36p, 44p, 52p, 66p, 68p and 76p are classified for each

and each prime

. The number of cubic

-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

-regular graphs of order 70p except for

= 31, 41.