In this paper, we design a pseudo-chaotic random number generator using fractional chaotic systems. A non-uniform grid calculation method is proposed and employed to numerically solve the fractional systems by introducing a skew-tent map to vary the step size of the grid. Greater chaoticity in terms of Lyapunov exponent has been achieved by applying the proposed approach to the solution calculation of the fractional chaotic Chen's and Lu's systems. By adopting the piecewise constant argument method, one 1D fractional generalized double-humped logistic system (FGDHL) is discretized. A fractional pseudo-chaotic random number generator (FPCRNG) has been proposed by performing XOR (exclusive-or) operations to the states of the fractional Chen's system, fractional Lu's system, and the FGDHL systems. The security analysis of the generator and the statistical experiment of a stream cipher implementing the proposed FPCRNG prove that the proposed structure is efficient and can be used in the cryptosystem design.