Long gamma-ray bursts (GRBs) have been linked to extreme core-collapse
supernovae from massive stars. Gravitational waves (GW) offer a probe of the
physics behind long GRBs. We investigate models of long-lived (~10-1000s) GW
emission associated with the accretion disk of a collapsed star or with its
protoneutron star remnant. Using data from LIGO's fifth science run, and GRB
triggers from the swift experiment, we perform a search for unmodeled
long-lived GW transients. Finding no evidence of GW emission, we place 90%
confidence level upper limits on the GW fluence at Earth from long GRBs for
three waveforms inspired by a model of GWs from accretion disk instabilities.
These limits range from F<3.5 ergs cm^-2 to $F<1200 ergs cm^-2, depending on
the GRB and on the model, allowing us to probe optimistic scenarios of GW
production out to distances as far as ~33 Mpc. Advanced detectors are expected
to achieve strain sensitivities 10x better than initial LIGO, potentially
allowing us to probe the engines of the nearest long GRBs.