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The quantum dynamics of a particle, with or without spin, in curved spacetime is analyzed from the Heisenberg picture, hence with a focus on the observables. It is based on a covariant Hamiltonian formulation. For the spin-zero particle, quantum geodesic equations are obtained as differential equations governing the evolution of the position observables. The formulation is not compatible with the usual approach based on the Schrödinger wavefunction representation, and has theoretically desirable features over the latter. Our approach suggests an alternative path to quantum gravitation based on the observables. While the classical, commutative, limit of the equations gives the geometric geodesic equation of real-number manifolds, the noncommutativity of the quantum observables gives a nontrivial form of the results. The (Weak) Equivalence Principle is maintained. Generalization to include spin is straightforward. Interesting conceptual implications and some results under black hole metrics will be discussed. (Class. Quantum Grav. 41, 085013 (2024) ; Chin. J. Phys. 96, 690-699; 97, 1293-1305 (2025); manuscript in preparation.)