We develop qutrit circuit models for discrete-time three-state quantum walks on Cayley graphs corresponding to Dihedral groups and the additive groups of integers modulo any positive integer . The proposed circuits comprise of elementary qutrit gates such as qutrit rotation gates, qutrit- gates and two-qutrit controlled- gates. First, we propose qutrit circuit representation of special unitary matrices of order three, and the block diagonal special unitary matrices with diagonal blocks, which correspond to multi-controlled gates and permutations of qutrit Toffoli gates. We show that one-layer qutrit circuit model need two-qutrit control gates and one-qutrit rotation gates for these quantum walks when . Finally we numerically simulate these circuits to mimic its performance such as time-averaged probability of finding the walker at any vertex on noisy quantum computers. The simulated results for the time-averaged probability distributions for noisy and noiseless walks are further compared using KL-divergence and total variation distance. These results show that noise in gates in the circuits significantly impacts the distributions than amplitude damping or phase damping errors.