The main problem in the one-point iterative methods with R-order of convergence at least three is the evaluation of the second derivative of the operator involved. Moreover, the convergence conditions required to prove the semilocal convergence of these methods are usually more restrictive than those required to Newton-like methods. In this chapter, we try to solve both problems. Thus, from the Chebyshev method, we construct families of iterative methods where the second derivative of the operator involved does not appear in the algorithms, but with the same R-order of convergence as the Chebyshev method, so that the operational cost of the new methods is lower than that of the Chebyshev method. We construct four modifications of the Chebyshev method that improve this method in some sense, and all of them are free of second derivatives. All studies are done in Banach spaces and using the technique based on recurrence relations described in Chap. 4 . Finally, the established theoretical results are illustrated with applications related to nonlinear Fredholm integral equations.