A search for gaugino pair production with a trilepton signature in the framework of R-parity violating supersymmetry via the couplings λ121, λ122, or λ133 is presented. The data, corresponding to an integrated luminosity of L≈360 pb−1, were collected from April 2002 to August 2004 with the D∅ detector at the Fermilab Tevatron Collider, at a center-of-mass energy of s=1.96 TeV. This analysis considers final states with three charged leptons with the flavor combinations eeℓ, μμℓ, and eeτ (ℓ=e or μ). No evidence for supersymmetry is found and limits at the 95% confidence level are set on the gaugino pair production cross section and lower bounds on the masses of the lightest neutralino and chargino are derived in two supersymmetric models.