We study the evolution of the cosmic dark energy by estimating the dark energy equation-of-state parameters with the recent observational data such as type Ia supernovae, cosmic microwave background radiation, baryon acoustic oscillations, and Hubble parameters. The Markov Chain Monte Carlo method is used to constrain the parameters of the standard ACDM model (ω=-1) and the ωCDM model in which the equation-of-state parameter ω is not equal to -1 or varies over time in both spatially flat and non-flat universes, respectively. As ωCDM models, we consider three cases : (i) XCDM with constant ω, (ii) Chevallier-Polarski-Linder parameterization with w linearly varying over the cosmic scale factor, (iii) piece-wise constant parameterization in which ω's are constant but with different values at two or three separate redshift intervals. In the spatially flat dark energy models we observe the tendency that ω>-1 in the early epoch (z≳0.5) and ω≈-1 in the late-time epoch (z≲0.5), which suggests the evolutionary characteristics of the dynamical dark energy. On the other hand, in the spatially non-flat model, it was shown that the dark energy equation-of-state parameter is statistically the same as that of the cosmological constant A (ω≈-1) over the whole redshift range and the spatially closed universe (Ωk, <0) is preferred. Our results are consistent with those of the previous studies.