We show that main-sequence stars in dense stellar cusps around massive black holes are likely to rotate at a significant fraction of the centrifugal breakup velocity due to spin-up by hyperbolic tidal encounters. We use realistic stellar structure models to calculate analytically the tidal spin-up in soft encounters, and extrapolate these results to close and penetrating collisions using smoothed particle hydrodynamics simulations. We find that the spin-up falls off only slowly with distance from the black hole because the increased tidal coupling in slower collisions at larger distances compensates for the decrease in the stellar density. We apply our results to the stars near the massive black hole in the Galactic Center. Over their lifetime, 1 M_o main sequence stars in the inner 0.3 pc of the Galactic Center are spun-up on average to 10% - 30% of the centrifugal breakup limit. Such rotation is 20 - 60 times higher than is usual for such stars and may affect their subsequent evolution and their observed properties.