We propose two novel, low-complexity techniques for beam alignment in time division duplexing (TDD) multiple-input multiple-output (MIMO) systems. The techniques are inspired by the power method, an iterative algorithm to determine eigenvalues and eigenvectors through repeated matrix multiplications and improve upon this simple idea by providing a better performance in the low-SNR regime. The first technique sequentially constructs a least-squares estimate of the channel matrix, which is then used to calculate the optimal beamformer/combiner pair. The second technique aims to mitigate the effect of additive noise by using a linear combination of the previously tried beams to calculate the next beam in the iteration. Simulation results provide insight on the performance of both algorithms in the presence of noise and compare them with similar techniques from the literature.