Let $$\mathbb R_{0, m+1}^{(s)}$$ be the space of s-vectors ( $$0\le s\le m+1$$ ) in the Clifford algebra $$\mathbb R_{0, m+1}$$ constructed over the quadratic vector space $$\mathbb R^{0, m+1}$$ , let $$r, p, q\in \mathbb N$$ with $$0\le r\le m+1$$ , $$0\le p\le q$$ and $$r+2q\le m+1$$ and let $$\mathbb R_{0, m+1}^{(r,p,q)}=\sum _{j=p}^q\bigoplus \mathbb R_{0, m+1}^{(r+2j)}$$ . Then a $$\mathbb R_{0, m+1}^{(r,p,q)}$$ -valued smooth function F defined in an open subset $$\Omega \subset \mathbb R^{m+1}$$ is said to satisfy the generalized Moisil–Teodorescu system of type (r, p, q) if $$\partial _x F=0$$ in $$\Omega $$ , where $$\partial _x$$ is the Dirac operator in $$\mathbb R^{m+1}$$ . To deal with the inhomogeneous generalized Moisil–Teodorescu systems $$\partial _x F=G$$ , with a $$\sum _{j=p}^{q} \bigoplus {\mathbb {R}}^{(r+2j-1)}_{0,m+1}$$ -valued continuous function G as a right-hand side, we embed the systems in an appropriate Clifford analysis setting. Necessary and sufficient conditions for the solvability of inhomogeneous systems are provided and its general solution described.