Motivated by recent experimental progress in the context of ultracold multicolor fermionic atoms in optical lattices, we have developed a method to exactly diagonalize the Heisenberg SU(N) Hamiltonian with several particles per site living in a fully symmetric or antisymmetric representation of SU(N). The method, based on the use of standard Young tableaux, takes advantage of the full SU(N) symmetry, allowing one to work directly in each irreducible representation of the global SU(N) group. Since the SU(N) singlet sector is often much smaller than the full Hilbert space, this enables one to reach much larger system sizes than with conventional exact diagonalizations. The method is applied to the study of Heisenberg chains in the symmetric representation with two and three particles per site up to N=10 and up to 20 sites. For the length scales accessible to this approach, all systems except the Haldane chain [SU(2) with two particles per site] appear to be gapless, and the central charge and scaling dimensions extracted from the results are consistent with a critical behavior in the SU(N) level k Wess-Zumino-Witten universality class, where k is the number of particles per site. These results point to the existence of a crossover between this universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical behavior in the SU(N) level 1 WZW universality class.