We consider the effect of classical phase fluctuations on the quasiparticle spectra of underdoped high-T-c cuprate superconductors in the pseudogap regime above T-c. We show that photoemission and tunneling spectroscopy data are well accounted for by a simple model in which mean-field d-wave quasiparticles are semiclassically coupled to supercurrents induced by fluctuating unbound vortex-antivortex pairs. We argue that the data imply that transverse phase fluctuations are important at temperatures above T-c, while longitudinal fluctuations are unimportant at all temperatures. [S0163-1829(98)05445-9].