We present a search for Higgs bosons decaying into b (b) over bar and produced in association with W bosons in p (p) over bar collisions at root s =1.96 TeV. This search uses 320 pb(-1) of the data set accumulated by the upgraded Collider Detector at Fermilab. Events are selected that have a high-transverse momentum electron or muon, missing transverse energy, and two jets, at least one of which is consistent with the hadronization of a b quark. Both the number of events and the dijet mass distribution are consistent with standard model background expectations, and we set 95% confidence level upper limits on the production cross section times branching ratio for the Higgs boson or any new particle with similar decay kinematics. These upper limits range from 10 pb for m(H)=110 GeV/c(2) to 3 pb for m(H)=150 GeV/c(2). RI Lancaster, Mark/C-1693-2008; Ruiz, Alberto/E-4473-2011; Robson, Aidan/G-1087-2011; De Cecco, Sandro/B-1016-2012; St.Denis, Richard/C-8997-2012; Prokoshin, Fedor/E-2795-2012; Azzi, Patrizia/H-5404-2012