We study mechanisms that allocate reviewers to papers in a fair and efficient manner. We model reviewer assignment as an instance of a fair allocation problem, presenting an extension of the classic round-robin mechanism, called Reviewer Round Robin (RRR). Round-robin mechanisms are a standard tool to ensure envy-free up to one item (EF1) allocations. However, fairness often comes at the cost of decreased efficiency. To overcome this challenge, we carefully select an approximately optimal round-robin order. Applying a relaxation of submodularity, γ-weak submodularity, we show that greedily inserting papers into an order yields a (1+γ²)-approximation to the maximum welfare attainable by our round-robin mechanism under any order. Our Greedy Reviewer Round Robin (GRRR) approach outputs highly efficient EF1 allocations for three real conference datasets, offering comparable performance to state-of-the-art paper assignment methods in fairness, efficiency, and runtime, while providing the only EF1 guarantee.