Soft iterative detection/decoding algorithms are fundamentally necessary for multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) adopted in the 3rd Generation Long Term Evolution-Advanced (LTE-A) in order to increase the capacity and achieve high data rates. However, their high performance critically requires log likelihood ratio (LLR) computations with prohibitive complexity. This challenge will be addressed in this paper. We first use the the assumption of Gaussian transmit symbols to show the equivalence among several existing algorithms. We next develop a non-Gaussian approximation for high-order constellations, which paves the way for interference cancellation based detectors. Based on both Gaussian and non-Gaussian approximations, we thus develop several capacity-achieving iterative MIMO-OFDM demodulation and decoding algorithms. To this end, we adopt Kbest algorithms to take advantage of both types of approximations and the list decoder. Unlike existing algorithms, our proposed K-best algorithms make use of the a priori probabilities to generate the list. Simulations of standard-compliant LTE systems demonstrate that the proposed algorithms outperform existing ones.