We study the neutralino sector of the left-right supersymmetric model. In addition to the possibilities available in the minimal supersymmetric model, the neutralino states can be superpartners of the U(1)(B-L) gauge boson, the neutral SU(2)(R) gauge boson, or of the Higgs triplets. We analyze neutralino masses and determine the parameter regions for which the lightest neutralino can be one of the new pure states. We then calculate the relic density of the dark matter for each of these states and impose the constraints coming from the rho parameter, the anomalous magnetic moment of the muon, b -> s gamma, as well as general supersymmetric mass bounds. The lightest neutralino can be the bino, or the right-wino, or the neutral triplet Higgsino, all of which have different couplings to the standard model particles from the usual neutralinos. A light bino satisfies all the experimental constraints and would be the preferred dark matter candidate for light supersymmetric scalar masses, while the right-wino would be favored by intermediate supersymmetric mass scales. The neutral triplet Higgs fermion satisfies the experimental bounds only in a small region of the parameter space, for intermediate to heavy supersymmetric scalar masses.