We report evidence for dynamically significant rotation in the globular cluster systems of two luminous Virgo dwarf ellipticals, VCC1261 and VCC1528. Including previous results for VCC1087, the globular cluster systems of all three Virgo dwarf ellipticals studied in detail to date exhibit v rot/σlos>1. Taking the rotation seen in the globular clusters as a maximal disk rotation, and accounting for the possible fading of any hypothetical progenitor galaxy, we find all three dwarf elliptical galaxies (dEs) lie on the r-band Tully-Fisher relation. We argue that these data support the hypothesis that luminous dEs are the remnants of transformed disk galaxies. We also obtained deep, longslit data for the stars in VCC1261 and VCC1528. Both these galaxies show rapid rotation in their inner regions, with spatial scales of ~0.5 kpc. These rotation velocities are surprisingly similar to those seen in the globular cluster systems. At larger radii, we see little rotation in the dEs themselves. Since our longslit data for Virgo dEs extend out to 1-2 effective radii (typical of deep observations), whereas the globular clusters extend out to 4-7 effective radii, we conclude that nondetections of rotation in many luminous dEs may simply be due to a lack of radial coverage in the stellar data, and that globular clusters represent singularly sensitive probes of the dynamics of dEs. Based on these data, we suggest that gas disks are significant sites of globular cluster formation in the early universe. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.