In order to cut all three pieces of timber into equal length logs, the length of the logs must be a factor of each of the three original lengths. The prime factors of 48 are $ 2^4\cdot3 $, those of 72 are $ 2^3\cdot3^2 $, and those of 40 are $ 2^3\cdot5 $. The greatest common factor of all three is $ 2^3=\boxed{8} $, so that is the greatest possible length the sawmill operator can cut.