Starting from a $3\times 3$ matrix spectral problem and the characteristic polynomial of the Lax matrix, we propose a trigonal curve, the associated meromorphic functions and three kinds of Abelian differentials. By discussing the asymptotic properties for the Baker-Akhiezer functions and their Riemann theta function expressions, we get quasi-periodic solutions of the three-component Burgers hierarchy. Finally, we straighten out the three-component Burgers flows.