We present the first model independent search for three-jet hadronic resonances within multijet events in sqrt(s) = 1.96 TeV pp collisions at the Fermilab Tevatron using the CDF II detector. Pair production of supersymmetric gluinos and squarks with hadronic R-parity violating decays is employed as an example of a new physics benchmark for this signature. Selection criteria based on the kinematic properties of an ensemble of jet combinations within each event help to extract signal from copious QCD background. No significant excess outside the top quark mass window is observed in data with an integrated luminosity of 3.2 fb(-1). We place 95% confidence level limits on the production cross section σ(pp → XX')×BR(gg → 3 jet + 3 jet) where X, X' = g, q, or q, with q, q → g + jet, as a function of gluino mass, in the range of 77 GeV/c2 to 240 GeV/c2.