  We develop a new representation for the integrals associated with Feynman
diagrams. This leads directly to a novel method for the numerical evaluation of
these integrals, which avoids the use of Monte Carlo techniques. Our approach
is based on based on the theory of generalized sinc ($\sin(x)/x$) functions,
from which we derive an approximation to the propagator that is expressed as an
infinite sum. When the propagators in the Feynman integrals are replaced with
the approximate form all integrals over internal momenta and vertices are
converted into Gaussians, which can be evaluated analytically. Performing the
Gaussians yields a multi-dimensional infinite sum which approximates the
corresponding Feynman integral. The difference between the exact result and
this approximation is set by an adjustable parameter, and can be made
arbitrarily small. We discuss the extraction of regularization independent
quantities and demonstrate, both in theory and practice, that these sums can be
evaluated quickly, even for third or fourth order diagrams. Lastly, we survey
strategies for numerically evaluating the multi-dimensional sums. We illustrate
the method with specific examples, including the the second order sunset
diagram from quartic scalar field theory, and several higher-order diagrams. In
this initial paper we focus upon scalar field theories in Euclidean spacetime,
but expect that this approach can be generalized to fields with spin.
