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In this paper we develop a fractional integro-differential operator calculus for Clifford-algebra valued functions. To do that we introduce fractional analogues of the Teodorescu and Cauchy-Bitsadze operators and we investigate some of their mapping properties. As a main result we prove a fractional Borel-Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge-type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann-Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.