Approximation properties of wavelet decompositions in the Orlicz spaces are investigated. Both dual wavelet frames and frame-like wavelet systems are considered. The order of approximation (in the sense of modular convergence) of wavelet frame decompositions satisfying a number of natural conditions is found for an arbitrary Orlicz space. For the Orlicz spaces satisfying Δ₂-condition, an error estimate providing approximation order of such wavelet expansions is given in terms of the Luxemburg norm. Similar results are obtained for appropriate frame-like systems under the assumption that an Orlicz space satisfies Δ^′-condition.