We present the results of a search for the rare flavor-changing neutral-current decays B -> pi l(+)l(-) (pi = pi(+), pi(0) and l = e, mu) and B-0 -> eta l(+)l(-) using a sample of e(+)e(-) -> Upsilon(4S) -> B (B) over bar decays corresponding to 428 fb(-1) of integrated luminosity collected by the BABAR detector. No significant signal is observed, and we set an upper limit on the isospin and lepton-flavor averaged branching fraction of B(B -> pi l(+)l(-)) eta l(+)l(-)) pi(+)e(+)e(-), B-0 -> pi(0)e(+)e(-), B-0 -> pi(+)mu(+)mu(-), B-0 -> pi(+)mu(+)mu(-), B-0 -> pi(0)mu(+)mu(-), B-0 -> eta e(+)e(-), and B-0 ->eta mu(+)mu(-)