The properties of pristine, free-standing graphene monolayers prepared by mechanical exfoliation of graphite are investigated. The graphene monolayers, suspended over open trenches, are examined by means of spatially resolved Raman spectroscopy of the G-, D-, and 2D-phonon modes. The G-mode phonons exhibit reduced energies (1580 cm(-1)) and increased widths (14 cm(-1)) compared to the response of graphene monolayers supported on the SiO(2)-covered substrate. From analysis of the G-mode Raman spectra, we deduce that the free-standing graphene monolayers are essentially undoped, with an upper bound of 2 x 10(11) cm(-2) for the residual carrier concentration. On the supported regions, significantly higher and spatially inhomogeneous doping is observed. The free-standing graphene monolayers show little local disorder, based on the very weak Raman D-mode response. The two-phonon 2D mode of the free-standing graphene monolayers is downshifted in frequency compared to that of the supported region of the samples and exhibits a narrowed, positively skewed line shape.