We investigate the influence of spin-orbit interaction on ballistic transport through chaotic cavities by using semiclassical methods. Our approach is based on the Landauer formalism [IBM J. Res. Dev. 1, 223 (1957); 32, 306 (1988)] and the Fisher-Lee relations [Phys. Rev. B 23, 6851 (1981)], appropriately generalized to spin-orbit interaction, and a semiclassical representation of Green's functions. We calculate conductance coefficients by exploiting ergodicity and mixing of suitably combined classical spin-orbit dynamics, and making use of the Sieber-Richter method [Phys. Scr., T T90, 128 (2001); Phys. Rev. Lett. 89, 206801 (2002)] and its most recent extensions. That way, we obtain weak antilocalization and confirm previous results obtained in the symplectic ensemble of random matrix theory.