With the AI boom, data center interconnects require higher symbol rates and throughput. Intensity modulation and direct detection (IM/DD) is a cost-effective candidate for short-reach optical links. Ring resonator-based modulators (RRMs) with broad bandwidth have emerged as a promising solution for high-speed IM/DD links. However, RRMs introduce nonlinear impairments, which can cause rate bottlenecks in IM/DD. In this work, we demonstrate up to 256 GBaud RRM-based optical amplification-free on-off keying links enabled by two NN-based equalizers. To the best of our knowledge, this work is the first RRM-based IM/DD demonstration of a 256 GBaud OOK link. We achieve bit-errorratio below the 7% overhead hard-decision forward error correction threshold after transmission over 100 m single-mode fiber (SMF) for all considered symbol rates. Our work outperforms recent state-of-the-art RRM-based IM/DD studies by applying NN equalization using a 42-GHz bandwidth RRM with a driving voltage of 2.7-V<inf>pp</inf>. The achieved performance and complexity analysis reported in this work are a key step towards future implementations of NN-based equalizers in hardware to enable high-symbol-rate IM/DD systems.