We develop a new extrapolation cascadic multigrid method ($\textrm{EXCMG}_{jcg}$), which makes it possible to solve 3D elliptic boundary value problems on rectangular domains of over 100 million unknowns on a desktop computer in half a minute. Test results from four examples with smooth and singular solutions, as well as a problem with variable coefficients are reported to show that $\textrm{EXCMG}_{jcg}$ has much better efficiency compared to the classical V-cycle and W-cycle multigrid methods. Finally, we present the reason why our EXCMG algorithms are highly efficient for solving these elliptic problems.