Transmission spectroscopy is one of the premier methods used to probe the temperature, composition, and cloud properties of exoplanet atmospheres. Recent studies have demonstrated that the multidimensional nature of exoplanet atmospheres -due to non-uniformities across the day-night transition and between the morning and evening terminators -can strongly influence transmission spectra. However, the computational demands of 3D radiative transfer techniques have precluded their usage within atmospheric retrievals. Here we introduce TRIDENT, a new 3D radiative transfer model which rapidly computes transmission spectra of exoplanet atmospheres with day-night, morning-evening, and vertical variations in temperature, chemical abundances, and cloud properties. We also derive a general equation for transmission spectra, accounting for 3D atmospheres, refraction, multiple scattering, ingress/egress, grazing transits, stellar heterogeneities, and nightside thermal emission. After introducing TRIDENT's linear algebra-based approach to 3D radiative transfer, we propose new parametric prescriptions for 3D temperature and abundance profiles and 3D clouds. We show that multidimensional transmission spectra exhibit two significant observational signatures: (i) day-night composition gradients alter the relative amplitudes of absorption features; and (ii) morning-evening composition gradients distort the peak-to-wing contrast of absorption features. Finally, we demonstrate that these signatures of multidimensional atmospheres incur residuals > 100 ppm compared to 1D models, rendering them potentially detectable with JWST. TRIDENT's rapid radiative transfer, coupled with parametric multidimensional atmospheres, unlocks the final barrier to 3D atmospheric retrievals.