  ABRIDGED: We introduce and analyze a method for testing statistical isotropy
and Gaussianity and apply it to the WMAP CMB foreground reduced, temperature
maps, and cross-channel difference maps. We divide the sky into regions of
varying size and shape and measure the first four moments of the one-point
distribution within these regions, and using their simulated spatial
distributions we test the statistical isotropy and Gaussianity hypotheses. By
randomly varying orientations of these regions, we sample the underlying CMB
field in a new manner, that offers a richer exploration of the data content,
and avoids possible biasing due to a single choice of sky division. The
statistical significance is assessed via comparison with realistic Monte-Carlo
simulations.
  We find the three-year WMAP maps to agree well with the isotropic, Gaussian
random field simulations as probed by regions corresponding to the angular
scales ranging from 6 deg to 30 deg at 68% confidence level. We report a
strong, anomalous (99.8% CL) dipole ``excess'' in the V band of the three-year
WMAP data and also in the V band of the WMAP five-year data (99.3% CL). We
notice the large scale hemispherical power asymmetry, and find that it is not
highly statistically significant in the WMAP three-year data (<~ 97%) at scales
l <= 40. The significance is even smaller if multipoles up to l=1024 are
considered (~90% CL). We give constraints on the amplitude of the
previously-proposed CMB dipole modulation field parameter. We easily detect the
residual foregrounds in cross-band difference maps at rms level <~ 7 \mu K (at
scales >~ 6 deg) and limit the systematical uncertainties to <~ 1.7 \mu K (at
scales >~ 30 deg).
