  We present a self-contained framework called Direct Integration of the
Relaxed Einstein Equations (DIRE) for calculating equations of motion and
gravitational radiation emission for isolated gravitating systems based on the
post-Newtonian approximation. We cast the Einstein equations into their
``relaxed'' form of a flat-spacetime wave equation together with a harmonic
gauge condition, and solve the equations formally as a retarded integral over
the past null cone of the field point (chosen to be within the near zone when
calculating equations of motion, and in the far zone when calculating
gravitational radiation). The ``inner'' part of this integral(within a sphere
of radius $\cal R \sim$ one gravitational wavelength) is approximated in a
slow-motion expansion using standard techniques; the ``outer'' part, extending
over the radiation zone, is evaluated using a null integration variable. We
show generally and explicitly that all contributions to the inner integrals
that depend on $\cal R$ cancel corresponding terms from the outer integrals,
and that the outer integrals converge at infinity, subject only to reasonable
assumptions about the past behavior of the source. The method cures defects
that plagued previous ``brute-force'' slow-motion approaches to motion and
gravitational radiation for isolated systems. We detail the procedure for
iterating the solutions in a weak-field, slow-motion approximation, and derive
expressions for the near-zone field through 3.5 post-Newtonian order in terms
of Poisson-like potentials.
