The $d$-Fibonacci digraphs $F(d,k)$, introduced here, have the number of vertices following generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their nice properties. For instance, $F(2,k)$ has diameter $d+k-2$ and is semi-pancyclic, that is, it has a cycle of every length between 1 and $\ell$, with $\ell\in\{2k-2,2k-1\}$. Moreover, it turns out that several other numbers of $F(d,k)$ (of closed $l$-walks, classes of vertices, etc.) also follow the same linear recurrences as the numbers of vertices of the $d$-Fibonacci digraphs.