We describe multiconfiguration linear-response (MCLR) approaches to the calculation of absolute photoionization cross sections. Three algorithms are described: (i) one in which the MCLR equations are used to derive the primitive spectrum of excitation energies and oscillator strengths; (ii) one in which the solutions of the MCLR equations are used to provide the even negative moments of the photoexcitation-photoionization spectrum; and (iii) one in which a pseudospectrum is obtained directly in the iterative procedure used to solve the MCLR equations. Either the primitive spectra (i), the moments (ii), or the pseudospectra (iii) are used as basic quantities in Stieltjes imaging to obtain the photoionization cross sections. Numerical demonstrations with large multiconfigurational self-consistent field reference spaces are given for the photoionization of HF, H2O, and Ne. Comparative calculations are performed in the random-phase approximation. Results are analyzed with respect to the fulfillment of gauge invariance, sum rules, basis-set completeness, and choice of correlating orbital spaces. Results for absolute photoionization cross sections from the MCLR algorithms agree very well with Stieltjes-imaging cross sections obtained from the semiempirically determined spectral moments of Meath and co-workers [Can. J. Phys. 55, 2080 (1977); 63, 1616 (1985)], and distinguish the raw experimental cross sections.