Inspired by papers by M.A. Bokhari, A. Qadir, and H. Al-Attas [On Gauss-type quadrature rules, Numer. Funct. Anal. Optim. 31 (2010), 1120-1134] and by M.R. Rapaic, T.B. Sekara, and V. Govedarica [A novel class of fractionally orthogonal quasi-polynomials and new fractional quadrature formulas, Appl. Math. Comput. 245 (2014), 206-219], in this paper we investigate a few types of orthogonal polynomials on finite intervals and derive the corresponding quadrature formulas of Gaussian type for efficient numerical computation of the left and right fractional Riemann-Liouville integrals. Several numerical examples are included to demonstrate the numerical efficiency of the proposed procedure.