This paper is concerned with output feedback control of discrete-time Lur'e systems with slope bounded nonlinearities that can be non-odd. Using FIR (Finite Impulse Response) Zames-Falb multipliers of any given order, stability conditions based on positive realness are proposed in terms of linear matrix inequalities and linear constraints that assure the ℓ 1 norm bound for the multiplier. Importantly, the proposed approach does not require line searches. Applying a relaxation on the stability and exploiting the fact that the matrices of the system appear affinely in the conditions, an iterative method is constructed to search for the FIR Zames-Falb multiplier certifying stability. An extension to cope with the design of a control law is also presented, dealing with state or output feedback gains that may include a term proportional to the output of the nonlinear block. Numerical examples illustrate the advantages of the proposed approach, that can be numerically more efficient than other techniques from the literature.