The Multilinear PageRank generalizes the classic Google PageRank to higher-order Markov chains by replacing the transition matrix with a stochastic tensor. This approach has been adopted for web ranking under complex link structures, and it supports higher-order graph representation learning, context-aware recommendation, and hypergraph analytics, all of which rely on tensor representations of multi-way interactions. The multilinear PageRank problem proves challenging to solve because it is a nonlinear tensor eigenproblem, and its solution is only guaranteed unique under a restricted parameter regime. Traditional solvers such as fixed-point iteration and Newton’s method suffer from slow convergence or expensive computational cost. In this work, we devise a modified shifted fixed-point iteration equipped with a periodic factor sequence to finely regulate convergence properties. Leveraging the Gerschgorin disc theorem, we further develop a Chebyshev acceleration framework tailored for the multilinear PageRank problem and establish its convergence theory. To alleviate the heavy computational overhead in large-scale scenarios, we construct an inexact variant of the proposed accelerated algorithm. Numerical tests confirm the superior performance and robustness of our method.