Models with large extra dimensions, like the ones proposed by Arkani-Hamed, Dimopoulos, and Dvali (ADD) on one hand, and Randall and Sundrum (RS) on the other hand, do predict gravitons within the reach of LHC. There are plenty of theoretical and experimental efforts directed into this search. For a recent review on ADD, see [1]. Even if you are not a professional, the introduction may give you an idea!
In the ADD model, the gravitons do not decay and therefore the signal would be missing energy, specifically $pp \to \text{jet} + \text{(missing energy)}$. There is a Standard Model (SM) background, $pp \to \text{jet} + Z$ where the $Z$ decays into a pair of neutrinos.
In the RS model, the gravitons decay into pairs of Standard Model particle. According to the Particle Data Group review, the most stringent limits come from decays into dilepton and diphoton. Here too there is a Standard Model background, the so-called Drell-Yan process $pp \to \text{jet} + Z^0/\gamma$ and then $Z/\gamma \to l^+ l^-$ for the dilepton.