This paper presents a design methodology for optimal energy allocation to estimate a random source using multiple wireless sensors equipped with energy harvesting technology. In this framework, multiple sensors observe a random process and then transmit an amplified uncoded analog version of the observed signal through Markovian fading wireless channels to a remote station. The sensors have access to an energy harvesting source, which is an everlasting but unreliable random energy source compared to conventional batteries with fixed energy storage. The remote station or so-called fusion centre estimates the realization of the random process by using a best linear unbiased estimator. The objective is to design optimal energy allocation policies at the sensor transmitters for minimizing total distortion over a finite-time horizon or a long term average distortion over an infinite-time horizon subject to energy harvesting constraints. This problem is formulated as a Markov decision process (MDP) based stochastic control problem and the optimal energy allocation policies are obtained by the use of dynamic programming techniques. Using the concept of submodularity, the structure of the optimal energy allocation policies is studied, which leads to an optimal threshold policy for binary energy allocation levels. Motivated by the excessive communication burden for the optimal control solutions where each sensor needs to know the channel gains and harvested energies of all other sensors, suboptimal decentralized strategies are developed where only statistical information about all other sensors' channel gains and harvested energies is required. Numerical simulation results are presented illustrating the performance of the optimal and suboptimal algorithms.