Abstract: We present the results of a search for pair production of a heavy toplike (t') quark decaying to Wq final states using data corresponding to an integrated luminosity of 5.6 fb(-1) collected by the CDF II detector in p (p) over bar collisions at root s = 1.96 TeV. We perform parallel searches for t' -> Wb and t' -> Wq (where q is a generic down-type quark) in events containing a lepton and four or more jets. By performing a fit to the two-dimensional distribution of total transverse energy versus reconstructed t' quark mass, we set upper limits on the t'(t) over bar' production cross section and exclude a standard model fourth-generation t' quark decaying to Wb (Wq) with mass below 358 (340) GeV/c(2) at 95% C. L.