The polaron problem is studied, on an infinite lattice, using the continuous-time path-integral quantum Monte Carlo scheme The method is based on the Feynman technique to analytically integrate out the phonon degrees of freedom. The transformed problem is that of a single electron with retarded self-interaction in imaginary time. The Metropolis algorithm is used to sample an ensemble of electron trajectories with twisted (rather than periodic) boundary conditions in imaginary time, which allows dynamic properties of the system to by measured directly. The method is numerically "exact", in the sense that there are no systematic errors due to finite system size, trotter decomposition or finite temperature The implementation of the algorithm in continuous imaginary time dramatically increases computational efficiency compared with the traditional discrete imaginary time algorithms.