Galaxy bispectrum is a promising probe of inflationary physics in the early Universe as a measure of primordial non-Gaussianity (PNG), whereas its signal-to-noise ratio is significantly affected by the mode coupling due to nonlinear gravitational growth. In this paper, we examine the standard reconstruction method of linear cosmic mass density fields from nonlinear galaxy density fields to decorrelate the covariance in redshift-space galaxy bispectra. In particular, we evaluate the covariance of the bispectrum for massive-galaxy-sized dark matter halos with reconstruction by using 4000 independent N-body simulations. Our results show that the bispectrum covariance for the postreconstructed field approaches the Gaussian prediction at scale of k <0.2 h Mpc(-1). We also verify the leading-order PNG-induced bispectrum is not affected by details of the reconstruction with perturbative theory. We then demonstrate the constraining power of the postreconstructed bispectrum for PNG at redshift of approximately 0.5. Further, we perform a Fisher analysis to make a forecast of PNG constraints by galaxy bispectra including anisotropic signals. Assuming a massive galaxy sample in the Sloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey, we find that the postreconstructed bispectrum can constrain the local, equilateral, and orthogonal types of PNG with Delta f(NL) similar to 13, 90, and 42, respectively, improving the constraints with the prereconstructed bispectrum by a factor of 1.3-3.2. In conclusion, the reconstruction plays an essential role in constraining various types of PNG signatures with a level of Delta f(NL) less than or similar to 1 from the galaxy bispectrum based on upcoming galaxy surveys.