In 2009, Cygnus X-3 (Cyg X-3) became the first microquasar to be detected in the GeV γ-ray regime, via the satellites Fermi and AGILE. The addition of this new band to the observational toolbox holds promise for building a more detailed understanding of the relativistic jets of this and other systems. We present a rich data set of radio, hard and soft X-ray, and γ-ray observations of Cyg X-3 made during a flaring episode in 2010 May. We detect a ~3 day softening and recovery of the X-ray emission, followed almost immediately by a ~1 Jy radio flare at 15 GHz, followed by a 4.3σ γ-ray flare (E > 100 MeV) ~1.5 days later. The radio sampling is sparse, but we use archival data to argue that it is unlikely the γ-ray flare was followed by any significant unobserved radio flares. In this case, the sequencing of the observed events is difficult to explain in a model in which the γ-ray emission is due to inverse Compton scattering of the companion star's radiation field. Our observations suggest that other mechanisms may also be responsible for γ-ray emission from Cyg X-3.