Dense retrieval models have predominantly been studied for English, where models have shown great success, due to the availability of human-labeled training pairs. However, there has been limited success for multilingual retrieval so far, as training data is uneven or scarcely available across multiple languages. Synthetic training data generation is promising (e.g., InPars or Promptagator), but has been investigated only for English. Therefore, to study model capabilities across both cross-lingual and monolingual retrieval tasks, we develop SWIM-IR, a synthetic retrieval training dataset containing 33 (high to very-low resource) languages for training multilingual dense retrieval models without requiring any human supervision. To construct SWIM-IR, we propose SAP (summarize-then-ask prompting), where the large language model (LLM) generates a textual summary prior to the query generation step. SAP assists the LLM in generating informative queries in the target language. Using SWIM-IR, we explore synthetic fine-tuning of multilingual dense retrieval models and evaluate them robustly on three retrieval benchmarks: XOR-Retrieve (cross-lingual), XTREME-UP (cross-lingual) and MIRACL (monolingual). Our models, called SWIM-X, are competitive with human-supervised dense retrieval models, e.g., mContriever, finding that SWIM-IR can cheaply substitute for expensive human-labeled retrieval training data.