We report on large scale finite-temperature Monte Carlo simulations of the classical 120 degrees or e(g) orbital-only model on the simple cubic lattice in three dimensions with a focus towards its critical properties. This model displays a continuous phase transition to an orbitally ordered phase. While the correlation length exponent nu approximate to 0.665 is close to the 3D XY value, the exponent eta approximate to 0.15 differs substantially from O(N) values. We also introduce a discrete variant of the e(g) model, called the e(g)-clock model, which is found to display the same set of exponents. Further, an emergent U( 1) symmetry is found at the critical point T-c, which persists for T < T-c below a crossover length scaling as Lambda similar to xi(a), with an unusually small a approximate to 1.3.