Measurement of laser-driven shock wave pressure in solid blocks of polymethyl methacrylate is demonstrated using fiber optic pressure probes. Three probes based on a fiber Fabry-Perot, fiber Bragg grating, and interferometric fiber tip sensor are tested and compared. Shock waves are generated using a high-power laser focused onto a thin foil target placed in close proximity to the test blocks. The fiber Fabry-Perot sensor appears capable of resolving the shock front with a rise time of 91 ns. The peak pressure is estimated, using a separate shadowgraphy measurement, to be 3.4 GPa.