In this paper, we investigate joint network and channel decoding algorithms for the multiple-access relay channel. We consider a realistic reference scenario with Rayleigh fading over all the wireless links, including the source-torelay channels. Our contribution is twofold: i) first, we develop the quasi-optimal joint network and channel decoder by taking into account possible errors over the source-torelay channels, and ii) second, we propose a low-complexity iterative joint network and channel decoding algorithm, which reduces the number of channel decoders with respect to state-of-the-art solutions. Our numerical results show that: i) in fully-interleaved Rayleigh fading channels, the proposed solution provides almost the same bit error probability as the quasi-optimal scheme but with a reduction in complexity of approximately the 65%, and ii) in Rayleigh block-fading channels, the proposed scheme yields almost the same bit error probability as state-of-the-art solutions but with a reduction in complexity of approximately the 30%.