Massive stars generally end their lives as neutron stars (NSs) or black holes (BHs), with NS formation typically occurring at the low-mass end and collapse to a BH more likely at the high-mass end. In an intermediate regime, with a mass range that depends on the uncertain details of rotation and mass loss during the star s life, an NS is initially formed, which then experiences fallback accretion and collapse to a BH. The electromagnetic consequence of such an event is not clear. Depending on the progenitor s structure, possibilities range from a long gamma-ray burst to a Type II supernova (which may or may not be jet powered) to a collapse with a weak electromagnetic signature. Gravitational waves (GWs) provide the exciting opportunity to peer through the envelope of a dying massive star and directly probe what is occurring inside. We explore whether fallback onto young NSs can be detected by ground-based interferometers. When the incoming material has sufficient angular momentum to form a disk, the accretion spins up the NS sufficiently to produce non-axisymmetric instabilities and gravitational radiation at frequencies of 700-2400 Hz for 30-3000 s until collapse to a BH occurs. Using a realistic excess cross-power search algorithm, we show that such events are detectable by Advanced LIGO out to 17 Mpc. From the rate of nearby core-collapse supernovae in the past five years, we estimate that there will be 1-2 events each year that are worth checking for fallback GWs. The observation of these unique GW signatures coincident with electromagnetic detections would identify the transient events that are associated with this channel of BH formation, while providing information about the protoneutron star progenitor.