In this article we present two types of nonlinear positivity-preserving finite volume (PPFV) schemes for a class of three-dimensional heat conduction equations on general polyhedral meshes. First, we present a new parameter selection strategy on the one-sided flux and establish a nonlinear PPFV scheme based on a two-point flux with higher efficiency. By comparing with the scheme proposed in [H. Xie, X. Xu, C. Zhai, H. Yong, Commun. Comput. Phys. 24 (2018) 1375–1408], our scheme avoids the assumption that the values of auxiliary unknowns are nonnegative, which makes our interpolation formulae suitable to be constructed by existing approaches with high accuracy and well robustness (e.g., the finite element method), thus enhancing the adaptability to distorted meshes with large deformations. Then we derive a linear multi-point flux involving combination coefficients and, via the Patankar trick, obtain another nonlinear PPFV scheme that is concise and easy to implement. The selection strategy of combination coefficients is also provided to improve the convergence behavior of the Picard procedure. Furthermore, the existence and positivity-preserving properties of these two nonlinear PPFV solutions are proved. Numerical experiments with the discontinuous diffusion scalar as well as discontinuous and anisotropic diffusion tensors are given to confirm our theoretical findings and demonstrate that our schemes both can achieve ideal-order accuracy even on severely distorted meshes.