Diffusion models are powerful priors for accelerated MRI reconstruction, but most current methods are posterior samplers, which return a random draw from the posterior. A reproducible and high-quality reconstruction often requires averaging many samples, resulting in a high computational cost. We introduce D-MOD, a fast algorithm that instead provides a posterior mode. It uses a minorize-maximize algorithm on a lower bound of the smoothed log-posterior, which provably increases the objective. Each iteration draws S noisy samples of the current estimate at the current noise level, denoises them with a pretrained diffusion model, and combines their average with the measured k-space through a conjugate-gradient (CG) solve. We anneal the noise levels, as the distance between the bound and its target decays quadratically with noise. On fastMRI brain (T1, T2, FLAIR), a single D-MOD reconstruction exceeds, in PSNR, the 10-sample finite-MMSE estimates of DPS, DAPS and SNORE-P and stays within 0.31 dB of the strongest baselines (DiffPIR, DDS). This approach thus provides a fast deterministic reconstruction algorithm (11-73 times faster than MMSE estimates), which is useful in a clinical setting.