We present multiwaveband observations of a well-selected sample of 28 quasars and two radio galaxies with flat radio spectra and strong millimeter-wave emission ( referred to here as FSRQ ’ s ) . The data are analyzed to determine the radio to infrared and X-ray to \gamma -ray properties of FSRQ ’ s and the relationships between them . Specifically , the synchrotron self-Compton ( SSC ) process is examined as a likely common radiation mechanism . For most sources , the broad band spectra are still incomplete , especially in the far-infrared and ultraviolet range . Therefore precise analysis , such as model-fitting of spectra is not usually possible . To compensate partially for this , we have taken a statistical approach , and examine the relationship between high and low energy emission by using the data set for the entire sample . We use very long baseline interferometry ( VLBI ) at 8.4 and 22 GHz — higher frequencies than those of previous surveys — in conjunction with nearly simultaneous radio to submillimeter-wave observations to determine the parameters of the synchrotron spectrum and to examine the compact angular structure of a subset of sources from our sample . These parameters are used to predict the SSC X-ray flux densities . Seven of thirty sources have predicted self-Compton X-ray flux densities well above the observed flux densities obtained with the ROSAT satellite unless one assumes that the radiating plasma experiences bulk relativistic motion directed toward the observer ’ s line of sight . Three of these seven sources are detected at \gamma -ray frequencies . Model spectra show that the X-rays are consistent with the first order SSC process , with the simultaneous multiwaveband spectrum of the quasar 0836+710 obtained in 1992 March being very well fit by SSC emission from a uniform , relativistically moving source . The \gamma -rays are not produced via second order self-Compton scattering , but rather by either first order self-Compton scattering or some other process . A comparison of the ROSAT X-ray flux densities and those obtained earlier with the Einstein Observatory show that several FSRQ ’ s are X-ray variables on timescales of about a decade . Several sources that were observed more than once with ROSAT also show variability on timescales of 1–2 yr , with the X-ray variability in these cases often associated with millimeter-wave variability and lower VLBI core-to-jet flux ratios . Detections at \gamma -ray energies also appear to be related to increases in the radio to millimeter-wave flux densities . Statistical analysis shows that the millimeter-wave and X-ray luminosities for the sample are strongly correlated , with a linear regression slope \sim 0.6 . The peak in the distribution of X-ray to millimeter spectral indices also indicates a strong connection between millimeter-wave and X-ray emission . Particularly interesting is a correlation between X-ray to millimeter spectral index and fraction of flux density contained in the VLBI core . This tendency toward higher X-ray fluxes from sources with stronger jet emission implies that the knots in the jet are prominent sources of X-rays .