We investigate the cosmological predictions of several f(T) models, with up to two parameters, at both the background and the perturbation levels. Using current cosmological observations (geometric supernovae type Ia, cosmic microwave background and baryonic acoustic oscillation and dynamical growth data) we impose constraints on the distortion parameter, which quantifies the deviation of these models from the concordance Lambda cosmology at the background level. In addition we constrain the growth index gamma predicted in the context of these models using the latest perturbation growth data in the context of three parametrizations for gamma. The evolution of the best fit effective Newton constant, which incorporates the f(T)-gravity effects, is also obtained along with the corresponding 1 sigma error regions. We show that all the viable parameter sectors of the f(T) gravity models considered practically reduce these models to Lambda CDM. Thus, the degrees of freedom that open up to Lambda CDM in the context of f(T) gravity models are not utilized by the cosmological data leading to an overall disfavor of these models.