This is what I managed to figure out via google. In "Decompositions into cycles I, Hamilton Decompositions" by Alspach, Bermond and Sotteau (google books link) of 1990, a regular graph $G$ with edge-set $E(G)$ has a *Hamilton decomposition* if either:

(i) $\text{deg}(G)=2d$ and $E(G)$ can be partitioned into $d$ Hamilton cycles, or

(ii) $\text{deg}(G)=2d+1$ and $E(G)$ can be partitioned into $d$ Hamilton cycles and a perfect matching.

Proposition 1 of that same paper claims that the $n$-cube has a Hamilton decomposition for all $n$. They considered the undirected graph of the hypercube, but it certainly answers your question about $D$ positively when $d$ (in the notation of your question) is even.

Their proof in the even case relies on the fact that the $2m$-cube can be written as a product $C_4\times\cdots\times C_4$ (where $C_4$ is the 4-cycle) and a corollary of a theorem of Aubert and Schneider's:

If $C$ is a cycle and $G$ is a 4-regular graph which is decomposable into two Hamilton cycles, then $C\times G$ can be decomposed into three Hamilton cycles.

I haven't had a chance to think through whether their construction helps in giving you a decomposition of your graph $D$ into Hamiltonian cycles when $d$ is odd. Their construction in that case relies on another paper by Alspach, Heinrich and Liu, "Orthogonal Factorizations of Graphs" which I didn't try to read yet.

It seems that a simple construction for these decompositions for the n-cube is still unknown. I found some discussion of $n=6,8$ in this paper of Okuda and Song.

There are also some slides of Alspach on Hamilton Decompositions which I found here.