The existence of Dark Matter (DM) in the form of Strongly Interacting Massive Particles (SIMPs) may be motivated by astrophysical observations that challenge the classical Cold DM scenario. Other observations greatly constrain, but do not completely exclude, the SIMP alternative. The signature of SIMPs at the LHC may consist of neutral, hadron-like, trackless jets produced in pairs. We show that the absence of charged content can provide a very efficient tool to suppress dijet backgrounds at the LHC, thus enhancing the sensitivity to a potential SIMP signal. We illustrate this using a simplified SIMP model and present a detailed feasibility study based on simulations, including a dedicated detector response parametrization. We evaluate the expected sensitivity to various signal scenarios and tentatively consider the exclusion limits on the SIMP elastic cross section with nucleons.