We report on a comprehensive all-sky search for periodic gravitational waves in the frequency band 100-1500 Hz and with a frequency time derivative in the range of [-1.18; +1.00] x 10(-8) Hz/s. Such a signal could be produced by a nearby spinning and slightly nonaxisymmetric isolated neutron star in our galaxy. This search uses the data from the initial LIGO sixth science run and covers a larger parameter space with respect to any past search. A Loosely Coherent detection pipeline was applied to follow up weak outliers in both Gaussian (95% recovery rate) and non-Gaussian (75% recovery rate) bands. No gravitational wave signals were observed, and upper limits were placed on their strength. Our smallest upper limit on worst-case (linearly polarized) strain amplitude h(0) is 9.7 x 10(-25) near 169 Hz, while at the high end of our frequency range we achieve a worst-case upper limit of 5.5 x 10(-24). Both cases refer to all sky locations and entire range of frequency derivative values.