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Although magnesium alloys deform extensively through shear strains and crystallographic re-orientations associated with the growth of twins, little is known about the strengthening mechanisms associated with this deformation mode. A crystal plasticity based phase field model for twinning is employed in this work to study the strengthening mechanisms resulting from the interaction between twin growth and precipitates. The full-field simulations reveal in great detail the pinning and de-pinning of a twin boundary at individual precipitates, resulting in a maximum resistance to twin growth when the precipitate is partially embedded in the twin. Furthermore, statistically representative precipitate distributions are used to systematically investigate the influence of key microstructural parameters such as precipitate orientation, volume fraction, size, and aspect ratio on the resistance to twin growth. The results indicate that the effective critical resolved shear stress (CRSS) for twin growth increases linearly with precipitate volume fraction and aspect ratio. For a constant volume fraction of precipitates, reduction of the precipitate size below a critical level produces a strong increase in the CRSS due to the Orowan-like strengthening mechanism between the twin interface and precipitates. Above this level the CRSS is size independent. The results are quantitatively and qualitatively comparable with experimental measurements and predictions of mean-field strengthening models. Based on the results, guidelines for the design of high strength magnesium alloys are discussed.