  There are various results that frame left-orderability of a group as a
geometric property. Indeed, the fundamental group of a 3-manifold is
left-orderable whenever the first Betti number is positive; in the case that
the first Betti number is zero this property is closely tied to the existence
of certain nice foliations. As a result, many large classes of 3-manifolds,
including knot complements, are known to have left-orderable fundamental group.
However, though the complement of a knot has left-orderable fundamental group,
the result of Dehn surgery is a closed 3-manifold that need not have this
property. We take this as motivation for the study of left-orderability in the
context of Dehn surgery, and establish a condition on peripheral elements that
must hold whenever a given Dehn surgery yields a manifold with left-orderable
fundamental group. This leads to a workable criterion used to determine when
sufficiently positive Dehn surgery produces manifolds with non-left-orderable
fundamental group. As examples we produce infinite families of hyperbolic knots
-- subsuming the (-2,3,q)-pretzel knots -- for which sufficiently positive
surgery always produces a manifold with non-left-orderable fundamental group.
Our examples are consistent with the observation that many (indeed, all known)
examples of L-spaces have non-left-orderable fundamental group, as the given
families of knots are hyperbolic L-space knots. Moreover, the behaviour of the
examples studied here is consistent with the property that sufficiently
positive surgery on an L-space knot always yields an L-space.
