The infrared vibrational-rotational spectra of the deuterated triatomic hydrogen molecular ions, H₂D⁺, HD₂⁺, and D₃⁺ have been observed with the Doppler-tuned ion-beam laser spectroscopic method with collision detection. Triatomic hydrogen molecular ions are produced in a coaxial electron-impact ion source. Next, the ions are accelerated and formed into a beam of several keV energy, which is then intercepted at a small angle by a frequency-stabilized CO laser beam. The energy of the ion beam is adjusted to Doppler-shift an ion transition into resonance with a nearby laser line. On resonance, the laser light stimulates the transition to take place. If the resonating states differ in population, the laser-induced transition produces a net population transfer. The occurrence of population transfer is detected by monitoring the transmission of the ion beam through a gas target after laser interaction. The transmission through the target is dependent upon the ion beam population distribution and, therefore, the laser-induced transition can be detected by detecting the change of the transmission of the ion beam. A mass analyzer before the target gas facilitates the mass identification of the observed transitions. We have measured 45 D₃⁺ transition frequencies, 9 H₂D⁺ transition frequencies, and 31 HD₂⁺ transition frequencies, all between 1650 and 2000 cm⁻¹, to better than ±0.0005 cm⁻¹ or ±0.3 ppm. The identifications of the quantum numbers are still in progress. This study should greatly help the search of H₃⁺ and H₂D⁺ ions in interstellar medium.