  Central to the development of any new theory is the investigation of the
observable consequences of the theory. In the search for quantum gravity,
research in phenomenology has been dominated by models violating Lorentz
invariance (LI) -- despite there being, at present, no evidence that LI is
violated. Causal set theory is a LI candidate theory of QG that seeks not to
quantise gravity as such, but rather to develop a new understanding of the
universe from which both GR and QM could arise separately. The key hypothesis
is that spacetime is a discrete partial order: a set of events where the
partial ordering is the physical causal ordering between the events. This
thesis investigates Lorentz invariant QG phenomenology motivated by the causal
set approach. Massive particles propagating in a discrete spacetime will
experience diffusion in both position and momentum in proper time. This thesis
considers this idea in more depth, providing a rigorous derivation of the
diffusion equation in terms of observable cosmic time. The diffusion behaviour
does not depend on any particular underlying particle model. Simulations of
three different models are conducted, revealing behaviour that matches the
diffusion equation despite limitations on the size of causal set simulated. The
effect of spacetime discreteness on the behaviour of massless particles is also
investigated. Diffusion equations in both affine time and cosmic time are
derived, and it is found that massless particles undergo diffusion and drift in
energy. Constraints are placed on the magnitudes of the drift and diffusion
parameters by considering the blackbody nature of the CMB. Spacetime
discreteness also has a potentially observable effect on photon polarisation.
For linearly polarised photons, underlying discreteness is found to cause a
rotation in polarisation angle and a suppression in overall polarisation.
