Covariant quantum mechanics (CQM) is a background-indipendent reformulation of quantum mechanics in which the time coordinate is treated as a dynamical degree of freedom, rather than an external evolution parameter. CQM was originally conceived and developed as a basic relational quantum formalism for discussing some conceptual issues associated with quantum gravity in a simplified setting. However, in recent times, a few papers have fruitfully employed it to incorporate quantum-gravity effects such as spacetime noncommutativity into simple phenomenological models. In this Ph.D thesis, I explore in more detail this possibility, providing a first systematic investigation of CQM as a tool for quantum-gravity phenomenology, rather than a toy model of full quantum gravity. In particular, starting from the ordinary CQM of a single relativistic particle, I build generalized models for the description of free quantum particles propagating on noncommutative or curved spacetimes. The present work is theoretical in character. I mainly focus on the development and characterization of a CQM-based framework suitable for dealing with generic spacetime noncommutativity or metric, and study simple examples.