Discrepancies between model temperature trends and observations in the tropics potentially involve two distinct problems: errors in sea surface temperature (SST) trends; and a vertical profile of trends in the troposphere that is too top-heavy, with excessive warming in the tropical upper troposphere (TUT) per unit warming at the surface. The latter issue is particularly troubling given the robust nature of the constraint in models provided by the moist adiabatic lapse rate. It is important to distinguish carefully between these two separate problems, the former associated with ocean heat uptake, feedbacks, and forcings that must be studied in a coupled model, while the latter is primarily associated with internal atmospheric dynamics that can be studied in atmosphere/land models running over prescribed SSTs. There is a long-standing contention that models overestimate TUT warming per unit SST warming as compared to Microwave Sounding Unit (MSU) data. Using GFDL's HiRAM atmospheric model at roughly 50km horizontal resolution, we show that the AGCM TUT temperature trends are highly sensitive to small differences in the SST data. We define TUT temperature to be a linear combination of MSU channels that minimizes the signal from the surface and the stratosphere, with maximum weight around 400hPa. For the period 1981-2008, GFDL's HIRAM model forced with HURRELL SST gives a tropical mean (20S-20N) TUT trend of 0.214K/decade whereas the same model forced with HadISST gives a trend of 0.166K/decade, in good agreement with the RSS (Remote Sensing Systems) MSU trend of 0.165K/decade. The tropical average SST trend difference between the HURRELL and HADiSST of 0.010K/decade is too small to explain the difference in model trends. However, tropical average temperatures are closely tied to deep convection and precipitation, which occur preferentially at locations of warmer than average SSTs. Trends in areas of highest tropical SST differ much more between the two datasets, and trends in precipitation-weighted SSTs differ between HadISST and HURRELL by 0.034 K/decade, which explains the TUT trend difference. Our work shows that tropical upper tropospheric warming in AGCMs may be very sensitive to subtle differences in SST datasets. A corollary is that AGCMs are an important tool for examining the consistency and accuracy of datasets. Further, using the HadISST dataset as a boundary condition for the HiRAM model, rather than HURRELL (as used in most CMIP5 AMIP simulations), we obtain better agreement with observations also for trends in Atlantic hurricane frequency over the satellite period.