We consider theories where the Standard Model (SM) neutrinos acquire masses through the seesaw mechanism at the weak scale. We show that in such a scenario, the requirement that any pre-existing baryon asymmetry, regardless of its origin, not be washed out leads to correlations between the pattern of SM neutrino masses and the spectrum of new particles at the weak scale, leading to definite predictions for the LHC. For type I seesaw models with a TeV scale Z' coupled to SM neutrinos, we find that for a normal neutrino mass hierarchy, at least one of the right-handed neutrinos must be `electrophobic', decaying with a strong preference into final states with muons and tauons rather than electrons. For inverted or quasi-degenerate mass patterns, on the other hand, we find upper bounds on the mass of at least one right-handed neutrino. In particular, for an inverted mass hierarchy, this bound is 1 TeV, while the corresponding upper limit in the quasi-degenerate case is 300 GeV. Similar results hold in type III seesaw models, albeit with somewhat more stringent bounds. For the Type II seesaw case with a weak scale SU(2) triplet Higgs, we again find that an interesting range of Higgs triplet masses is disallowed by these considerations.