No theoretical predictions exist for the concentration dependence of long-time self-diffusion coefficients of rod-shaped Brownian particles with a finite aspect ratio. The reason for this is that the relevant Smoluchowski equation is extremely complicated and cannot be solved explicitly, even on the two-particle level. We present an alternative approach where the Smoluchowski equation is solved in approximation by a variational method. The variational principle is applied to calculate the dependence of the long-time translational self-diffusion coefficient of spherocylinders with hard-core interaction to leading order in concentration, with the neglect of hydrodynamic interactions, up to aspect ratios of 30. The first order in concentration coefficient alpha is found to depend on the aspect ratio p as alpha = 2 + 10/32(p - 1) + 1/53(p - 1)(2).