  Trimers are defined as two adjacent edges on a graph. We study the quantum
states obtained as equal-weight superpositions of all trimer coverings of a
lattice, with the constraint of having a trimer on each vertex: the so-called
trimer resonating-valence-bond (tRVB) states. Exploiting their tensor network
representation, we show that these states can host $\mathbb{Z}_3$ topological
order or can be gapless liquids with $\mathrm{U}(1) \times \mathrm{U}(1)$ local
symmetry. We prove that this continuous symmetry emerges whenever the lattice
can be tripartite such that each trimer covers all the three sublattices. In
the gapped case, we demonstrate the stability of topological order against
dilution of maximal trimer coverings, which is relevant for realistic models
where the density of trimers can fluctuate. Furthermore, we clarify the
connection between gapped tRVB states and $\mathbb{Z}_3$ lattice gauge theories
by smoothly connecting the former to the $\mathbb{Z}_3$ toric code, and discuss
the non-local excitations on top of tRVB states. Finally, we analyze via exact
diagonalization the zero-temperature phase diagram of a diluted trimer model on
the square lattice and demonstrate that the ground state exhibits topological
properties in a narrow region in parameter space. We show that a similar model
can be implemented in Rydberg atom arrays exploiting the blockade effect. We
investigate dynamical preparation schemes in this setup and provide a viable
route for probing experimentally $\mathbb{Z}_3$ quantum spin liquids.
