  We present a new, highly efficient yet accurate approximation for the Green's
functions of dressed particles, using the Holstein polaron as an example.
Instead of summing a subclass of diagrams (e.g. the non-crossed ones, in the
self-consistent Born approximation (SCBA)), we sum all the diagrams, but with
each diagram averaged over its free propagators' momenta. The resulting Green's
function satisfies exactly the first six spectral weight sum rules. All higher
sum rules are satisfied with great accuracy, becoming asymptotically exact for
coupling both much larger and much smaller than the free particle bandwidth.
Possible generalizations to other models are also discussed.
