Using a cohomological obstruction, we construct examples of absolutely simple adjoint classical groups of type 2A_n with n equivalent to 3 mod 4, C_n or 1D_n with n equivalent to 0 mod 4, which are not R-trivial hence not stably rational.
Berhuy, G., Monsurrò, M., & Tignol, J.-P. (2004). Cohomological invariants and R-triviality of adjoint classical groups. Mathematische Zeitschrift, 248(2), 313-323. https://doi.org/10.1007/s00209-003-0611-2 (Original work published 2004)