What was done
The authors applied the TEX86 (tetraether index of 86 carbon atoms) paleotemperature proxy -- which is linearly correlated with lake surface temperature (Kim et al., 2008) -- to sediment cores retrieved from Lake Tanganyika, which they analyzed at an average time step of 300 years.

What was learned
First of all, Tierney et al. note that both the magnitude and timing of the temperature rise observed at the termination of the last glacial period at Lake Tanganyika "are similar to those of nearby Lake Malawi," which indicates that their TEX86 record "captures regional temperature change in tropical southeast Africa during deglaciation." Second, they report that "Tanganyika lake surface temperature at the end of the Last Glacial Maximum follows rising Northern Hemisphere summer insolation," which they describe as "a potential trigger for deglaciation." And, third, they note that the initiation of the deglacial warming in southeast Africa "is consistent with rising temperatures at ~20,000 years B.P. in Antarctica," but they add that the deglacial warming "leads the deglacial CO2 rise recorded in Antarctic ice cores by about 3,000 years," which they describe as "a difference that is outside the chronological errors of the ice core and the lake surface temperature records."

What it means
In light of these several well-documented observations, Tierney et al. were forced -- by the power of simple logic -- to conclude that "increasing greenhouse gas concentrations are therefore not responsible for the initial transmission of warming from the high latitudes to the southeast African tropics." And if rising atmospheric CO2 concentrations were not responsible for the initial 3,000 years of post-glacial warming (since the air's CO2 content was essentially constant over this period), there is no compelling reason to believe they were responsible for any of the subsequent warming (when the air's CO2 content was in a rising mode). By analogy, therefore, there is also no compelling reason to believe that the more recent historical rise in the air's CO2 concentration has been responsible for any of the post-Little Ice Age warming of the globe.

Reference
Kim, J.-H., Schouten, S., Hopmans, E.C., Donner, B. and Sinninghe Damste, J.S. 2008. Global sediment core-top calibration of the TEX86 paleothermometer in the ocean. Geochimica et Cosmochimica Acta 72: 1154-1173.