Wireless links use adaptive modulation and coding (AMC) to optimize data transmission over a dynamic channel. Traditional AMC schemes rely on simple heuristics to track the instantaneous channel state. While attractive for their low implementation and operational complexity, these schemes are known to be suboptimal in a large range of operating environments. Further, several such schemes require careful parameter tuning, which can be both expensive and error-prone. In this paper, we propose latent Thompson sampling (LTS) for AMC, which efficiently tracks the wireless channel by modeling a latent, low-dimensional, channel state. LTS features both a low computational complexity and fast learning dynamics, and requires minimal tuning effort. We evaluate LTS in stationary as well as fading wireless channels, where LTS improves the link throughput by up to 100% compared to state-of-the-art schemes.