Polymers consisting of rigid segments connected by flexible joints (needle chains) constitute an important class of biopolymers. Using kinetic theory as a starting point, we first derive the generalized coordinate-space diffusion (Fokker-Planck) equation for the needle chain polymer model. Next, the equivalent generalized coordinate Ito stochastic differential equation is established. Nonlinear transformations of variables finally yield a stochastic differential equation for the needle chain spatial coordinates in the laboratory coordinate system where the coefficients are expressed in terms of the chain constraint conditions. This latter equation constitutes the basis for our needle chain Brownian dynamics (BD) algorithm. The used needle chain model includes needle translation-translation and rotation-rotation hydrodynamic interactions, a homogeneous solvent flow field, external forces, excluded volume effects, and bending and twisting stiffness between nearest neighbor segments. For this chain model we find that by proper generalization of the involved parameters the mathematical analysis of the polymer dynamics, in great detail, maps onto the analysis of the bead-rod-spring polymer chain model with constraints presented by Ottinger in Phys. Rev. E 50, 2696 (1994). Preliminary numerical simulation data show that for a three segment needle chain, with needle axial ratio equal to five, our new needle chain BD algorithm is, in general, more than about 10(3) times more efficient than the bead-spring polymer chain BD algorithm commonly used as an approximation for studies of such polymer chains, This efficiency ratio increases asymptotically proportional to approximately the fourth power of the needle axial ratio. In addition to this major gain in efficiency, the needle chain model for segmented polymers, in general, incorporates a more realistic hydrodynamic description of the individual segments and, in particular, the joints between the segments than the bead-rod-spring models.