The first direct detection of neutron-star– black-hole binaries will likely be made with gravitational-wave observatories. Advanced LIGO and Advanced Virgo will be able to observe neutron-star– black-hole mergers at a maximum distance of 900 Mpc. To achieve this sensitivity, gravitational-wave searches will rely on using a bank of filter waveforms that accurately model the expected gravitational-wave signal. The emitted signal will depend on the masses of the black hole and the neutron star and also the angular momentum of both components. The angular momentum of the black hole is expected to be comparable to the orbital angular momentum when the system is emitting gravitational waves in Advanced LIGO’s and Advanced Virgo’s sensitive band. This angular momentum will affect the dynamics of the inspiralling system and alter the phase evolution of the emitted gravitational-wave signal. In addition, if the black hole’s angular momentum is not aligned with the orbital angular momentum, it will cause the orbital plane of the system to precess. In this work we demonstrate that if the effect of the black hole’s angular momentum is neglected in the waveform models used in gravitational-wave searches, the detection rate of (10 + 1.4) M⊙ neutron-star– black-hole systems with isotropic spin distributions would be reduced by 33%–37% in comparison to a hypothetical perfect search at a fixed signal-to-noise ratio threshold. The error in this measurement is due to uncertainty in the post-Newtonian approximations that are used to model the gravitational-wave signal of neutron-star– black-hole inspiralling binaries. We describe a new method for creating a bank of filter waveforms where the black hole has nonzero angular momentum that is aligned with the orbital angular momentum. With this bank we find that the detection rate of (10 + 1.4) M⊙ neutron-star– black-hole systems would be reduced by 26%–33%. Systems that will not be detected are ones where the precession of the orbital plane causes the gravitational-wave signal to match poorly with nonprecessing filter waveforms. We identify the regions of parameter space where such systems occur and suggest methods for searching for highly precessing neutron-star– black-hole binaries.