The question is "no" in general, as Gjergji pointed out. The question of when the answer is "yes" is difficult and only partial results are known. It is conjectured that having a 1-factor is sufficient. A recent paper that discusses the issue is this paper of Odile Favaron, François Genest and Mekkia Kouider. Here is the abstract:
Kotzig asked in 1979 what are necessary and sufficient conditions
for a $d$-regular simple graph to admit a decomposition into paths
of length $d$ for odd $d\gt 3$. For cubic graphs, the existence of a 1-factor is
both necessary and sufficient. Even more, each 1-factor is extendable to a
decomposition of the graph into paths of length 3 where the middle edges
of the paths coincide with the 1-factor. We conjecture that existence of a
1-factor is indeed a sufficient condition for Kotzig’s problem. For general
odd regular graphs, most 1-factors appear to be extendable and we show
that for the family of simple 5-regular graphs with no cycles of length
4, all 1-factors are extendable. However, for $d\gt 3$ we found infinite families of $d$-regular simple graphs with non-extendable 1-factors. Few authors
have studied the decompositions of general regular graphs. We present
examples and open problems; in particular, we conjecture that in planar
5-regular graphs all 1-factors are extendable.
Finally, I should mention that I'm not a specialist in this area so my assertion that the problem is still unsolved is not gospel.