This paper studies optimal linear power control for battery-limited energy harvesting communications. It provides a systematic analysis of linear power control policies, covering the greedy and fixed-fraction policies as special cases. Three optimality notions are introduced: the maximin optimal linear policy for a given battery capacity and mean-to-capacity ratio (MCR), and two capacity-agnostic policies that minimize the nominal additive gap and maximize the nominal multiplicative factor, respectively. Except the capacity-agnostic additive-gap optimal linear policy, which coincides with the fixed-fraction policy, the other two optimal linear policies are novel and constitute the main contributions of this paper. It is shown, among others, that the worst nominal multiplicative factor for both novel policies is approximately 0.6530, a substantial improvement over the fixed-fraction policy's value of 0.5. Simulations show that under quasi-static fading, the maximin optimal linear policy performs comparable to the maximin optimal policy (the top-performing policy), while the capacity-agnostic multiplicative-factor optimal linear policy performs slightly worse; nevertheless, both novel policies significantly outperform the fixed-fraction policy in the low-to-medium signal-to-noise ratio (SNR) regime. Moreover, this paper also investigates the optimality of the greedy policy for certain families of energy-arrival distributions, and establishes the tightest semi-universal bounds on the battery-capacity threshold for greedy optimality.