In this paper, we study the optimum estimation of a continuous-time random process by using discrete-time samples taken by a sensor powered by energy harvesting power sources. The system employs a best-effort sensing scheme to cope with the stochastic nature of the energy harvesting sources. The best-effort sensing scheme defines a set of equally-spaced candidate sensing instants, and the sensor performs sensing at a given candidate sensing instant if there is sufficient energy available, and remains silent otherwise. It is shown through asymptotic analysis that when the energy harvesting rate is strictly less than the energy consumption rate, there is a non-negligible percentage of silent symbols due to energy outage. For a given average energy harvesting rate, a larger sampling period means a smaller energy outage probability and/or more energy per sample, but a weaker temporal correlation between two adjacent samples. Such a tradeoff relationship is captured by developing a closed-form expression of the estimation MSE, which analytically identifies the interactions among the various system parameters, such as the ratio between the energy harvesting rate and energy consumption rate, the sampling period, and the energy allocation between sensing and transmission. It is shown through theoretical analysis that the optimum performance can be achieved by adjusting the sampling period and sampling energy such that the average energy harvesting rate is equal to the average consumption rate.