Locating the gamma-ray emission sites in blazar jets is a long-standing and highly controversial issue. We investigate jointly several constraints on the distance scale r and Lorentz factor Gamma of the gamma-ray emitting regions in luminous blazars (primarily flat spectrum radio quasars, FSRQs). Working in the framework of one-zone external radiation Comptonization (ERC) models, we perform a parameter space study for several representative cases of actual gamma-ray flares in their multiwavelength context. We find a particularly useful combination of three constraints: from an upper limit on the collimation parameter Gamma*theta <~ 1, from an upper limit on the synchrotron self-Compton (SSC) luminosity L_SSC <~ L_X, and from an upper limit on the efficient cooling photon energy E_cool,obs <~ 100 MeV. These three constraints are particularly strong for sources with low accretion disk luminosity L_d. The commonly used intrinsic pair-production opacity constraint on Gamma is usually much weaker than the SSC constraint. The SSC and cooling constraints provide a robust lower limit on the collimation parameter Gamma*theta >~ 0.1 - 0.7. Typical values of r corresponding to moderate values of Gamma ~ 20 are in the range 0.1 - 1 pc, and are determined primarily by the observed variability time scale t_var,obs. Alternative scenarios motivated by the observed gamma-ray/mm connection, in which gamma-ray flares of t_var,obs ~ a few days are located at r ~ 10 pc, are in conflict with both the SSC and cooling constraints. Moreover, we use a simple light travel time argument to point out that the gamma-ray/mm connection does not provide a significant constraint on the location of gamma-ray flares. We argue that spine-sheath models of the jet structure do not offer a plausible alternative to external radiation fields at large distances, however, an extended broad-line region is an idea worth exploring.
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