We present a numerical study of the localized transverse magnetic (TM) defect modes in a two-dimensional, triangular-lattice photonic crystal. The sample consists of an array of circular, air cylinders in a dielectric medium (GaAs). The defect modes were calculated by using a parallel version of the finite-difference time-domain method on the Yee mesh. To validate our computations the results for the transverse electric case were checked against experimental results and the numerical results using a different method. We study the spatial symmetry for TM modes, obtained by changing the dipole excitation frequency. Also, we vary the defect-cylinder radius to tune the resonant frequency across the band gap. The TM mode is found to be highly localized at the defect in the photonic lattice.