In this paper, we propose a new Monte Carlo radiative transport (MCRT) scheme, which is based completely on the Neumann series solution of the Fredholm integral equation. This scheme indicates that the essence of MCRT is the calculation of infinite terms of multiple integrals in the Neumann solution simultaneously. Under this perspective, we redescribe the MCRT procedure systematically, in which the main work amounts to choosing an associated probability distribution function for a set of random variables and the corresponding unbiased estimation functions. We select a relatively optimal estimation procedure that has a lower variance from an infinite number of possible choices, such as term-by-term estimation. In this scheme, MCRT can be regarded as a pure problem of integral evaluation, rather than as the tracing of random-walking photons. Keeping this in mind, one can avert some subtle intuitive mistakes. In addition, the d functions in these integrals can be eliminated in advance by integrating them out directly. This fact, together with the optimal chosen random variables, can remarkably improve the Monte Carlo (MC) computational efficiency and accuracy, especially in systems with axial or spherical symmetry. An MCRT code, Lemon (Linear integral Equations' Monte carlo solver based On the Neumann solution; the code is available on the GitHub codebase at https://github.com/yangxiaolinyn/Lemon, and version 2.0 is archived on Zenodo at https://doi.org/10.5281/zenodo.4686355), has been developed completely based on this scheme. Finally, we intend to verify the validation of Lemon; a suite of test problems mainly restricted to a flat spacetime has been reproduced, and the corresponding results are illustrated in detail.