Monte Carlo Tree Search (MCTS), which leverages Upper Confidence Bound for Trees (UCTs) to balance exploration and exploitation through randomized sampling, is instrumental to solving complex planning problems. However, for multi-agent planning, MCTS is confronted with a large combinatorial action space that often grows exponentially with the number of agents. As a result, the branching factor of MCTS during tree expansion also increases exponentially, making it very difficult to efficiently explore and exploit during tree search. To this end, we propose MALinZero, a new approach to leverage low-dimensional representational structures on joint-action returns and enable efficient MCTS in complex multi-agent planning. Our solution can be viewed as projecting the joint-action returns into the low-dimensional space representable using a contextual linear bandit problem formulation. We solve the contextual linear bandit problem with convex and $μ$-smooth loss functions -- in order to place more importance on better joint actions and mitigate potential representational limitations -- and derive a linear Upper Confidence Bound applied to trees (LinUCT) to enable novel multi-agent exploration and exploitation in the low-dimensional space. We analyze the regret of MALinZero for low-dimensional reward functions and propose an $(1-\tfrac1e)$-approximation algorithm for the joint action selection by maximizing a sub-modular objective. MALinZero demonstrates state-of-the-art performance on multi-agent benchmarks such as matrix games, SMAC, and SMACv2, outperforming both model-based and model-free multi-agent reinforcement learning baselines with faster learning speed and better performance.