We test the performance of the Monte Carlo renormalization method in the context of the Ising model on atriangular lattice. We apply a block-spin transformation which allows for an adjustable parameter so that the transformation can be optimized. This optimization purportedly brings the fixed point of the transformation to a location where the corrections to scaling vanish. To this purpose we determine corrections to scaling of the triangular Ising model with nearest- and next-nearest-neighbor interactions by means of transfer-matrix calculationsand finite-size scaling. We find that the leading correction to scaling just vanishes for the nearestneighbormodel. However, the fixed point of the commonly used majority-rule block-spin transformation appears to lie well away from the nearest-neighbor critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because the standard assumptions of real-space renormalizationimply that corrections to scaling vanish at the fixed point. We avoid this inconsistency by means of theoptimized transformation which shifts the fixed point back to the vicinity of the nearest-neighbor criticalHamiltonian. The results of the optimized transformation in terms of the Ising critical exponents are moreaccurate than those obtained with the majority rule.