In this work, we present a new path-integral theory to calculate the ARPES of e=ph coupled systems. Our main purpose of this paper is to clarify the impact of e-ph interaction on the lineshape of ARPES. we consider a many-electron system coupled with Einstein phonons, which is often called Holstein model. This model has been studied extensivery, with various interests ranging from the competition between metallic, CDW and SC phases, to the energy gap opening, by perturbative and non-perturbative methods. In our numerical calculation, the path-integral is performed by the quantum Monte Carlo (QMC) simulation. Thus it is completely free from any other approximations. Give the QMC data of imaginary time Green's function, we reproduce the spectral function by the analytic continuation. By this means, we systematically study the spectral properties of the e-ph coupled systems based on the one-dimensional and two-dimensional (2D) Holstein models under various conditions. We find the band structure is greatly modified by the multiple scattering effect of electron with phonons, even if the whole system is still metallic and the e-ph coupling strength is intermediate. Around the band bottom, the spectrum takes a broad Gaussian, indicating the electron in this state is nearly localized and incoherent. While near the E