This article deals with the discretization of hybrid dimensional Darcy flows in fractured porous media. These models couple the flow in the fractures represented as surfaces of codimension one with the flow in the surrounding matrix. The convergence analysis is carried out in the framework of Gradient schemes which accounts for a large family of conforming and nonconforming discretizations. The Vertex Approximate Gradient (VAG) scheme and the Hybrid Finite Volume (HFV) scheme are extended to such models and are shown to verify the Gradient scheme framework. Our theoretical results are confirmed by numerical experiments performed on tetrahedral, Cartesian and hexahedral meshes in heterogeneous isotropic and anisotropic porous media.