Computational homogenization is a common tool for reducing the experimental effort required to characterize heterogeneous materials. Computational methods using the Fast Fourier Transform (FFT) to apply a preconditioner have proven to be highly effective. The original approach of Moulinec and Suquet works for periodic boundary conditions only. Handling Neumann or Dirichlet boundary conditions requires significant modifications of the computational scheme. Alternatively, these boundary conditions may be imposed by replacing the discrete Fourier transform with a discrete sine or cosine transform. FFT-based methods perform favorably because the simple representation of the preconditioner in Fourier space allows for an efficent computation of its action. In contrast, an application of the preconditioner in real space simplifies imposing the boundary conditions but amounts to the inversion of and multiplication with a large (nonetheless sparse) matrix with a high degree of redundant entries. Low-rank approaches -- such as the Tensor Train (TT) format and its extension, the Quantics Tensor Train (QTT) format -- enable a redundancy-free approximation and inversion of matrices, therefore making a real-space application of the preconditioner feasible. In this contribution, we investigate the efficacy of computational homogenization of heat conduction with a TT-based preconditioner on large-scale microstructures. For this purpose, we approximate Green's operator, i.e., the inverse of the finite difference Laplacian, in the QTT format.  We use the operator as a preconditioner for Lippmann-Schwinger-type solvers. In doing this, we apply periodic, Neumann or Dirichlet boundary conditions. We compare our work with an approach based on discrete sine and cosine transform and evaluate the performance and convergence of the TT-based solver. We highlight the advantages and disadvantages of our approach.