Slot-filling and intent detection are well-established tasks in Conversational AI. However, current large-scale benchmarks for these tasks often exclude evaluations of low-resource languages and rely on translations from English benchmarks, thereby predominantly reflecting Western-centric concepts. In this paper, we introduce Injongo -- a multicultural, open-source benchmark dataset for 16 African languages with utterances generated by native speakers across diverse domains, including banking, travel, home, and dining. Through extensive experiments, we benchmark the fine-tuning multilingual transformer models and the prompting large language models (LLMs), and show the advantage of leveraging African-cultural utterances over Western-centric utterances for improving cross-lingual transfer from the English language. Experimental results reveal that current LLMs struggle with the slot-filling task, with GPT-4o achieving an average performance of 26 F1-score. In contrast, intent detection performance is notably better, with an average accuracy of 70.6%, though it still falls behind the fine-tuning baselines. Compared to the English language, GPT-4o and fine-tuning baselines perform similarly on intent detection, achieving an accuracy of approximately 81%. Our findings suggest that the performance of LLMs is still behind for many low-resource African languages, and more work is needed to further improve their downstream performance.