One of the products of tryglyceride breakdown is glycerol; glycerol enters the cellular respiration pathway by being converted to glucose (in the liver) and then undergoing glycolysis.

I believe that the above statement is true. 
Triglycerides cannot pass through cell membranes freely. Special enzymes on the walls of blood vessels called lipoprotein lipases must break down triglycerides into free fatty acids and glycerol. Glycerol then enters the glycolytic pathway as dihydroxyacetone phosphate. Odd chain fatty acids are converted to propionyl-CoA, which can enter through mitochondrial succinyl-CoA.