  Models of type I seesaw can be implemented at the electroweak scale in a
natural way provided that the heavy neutrino singlets are quasi-Dirac
particles. In such case, their contribution to light neutrino masses has the
suppression of a small lepton number violating parameter, so that light
neutrino masses can arise naturally even if the seesaw scale is low and the
heavy neutrino mixing is large. We implement the same mechanism with fermionic
triplets in type III seesaw, deriving the interactions of the new quasi-Dirac
neutrinos and heavy charged leptons with the SM fermions. We then study the
observability of heavy Dirac neutrino singlets (seesaw I) and triplets (seesaw
III) at LHC. Contrarily to common wisdom, we find that heavy Dirac neutrino
singlets with a mass around 100 GeV are observable at the 5 sigma level with a
luminosity of 13 fb^-1. Indeed, in the final state with three charged leptons
l+- l+- l-+, not previously considered, Dirac neutrino signals can be
relatively large and backgrounds are small. In the triplet case, heavy
neutrinos can be discovered with a luminosity of 1.5 fb^-1 for a mass of 300
GeV in the same channel.
