We propose a mechanism to generate primordial black holes (PBHs) that is independent of cosmological inflation and occurs slightly below the QCD phase transition. Our setup relies on the collapse of long-lived string-domain wall networks and is naturally realized in QCD axion models with domain wall number N_{DW}>1 and Peccei-Quinn symmetry broken after inflation. In our framework, dark matter is mostly composed of axions in the meV mass range along with a small fraction, Ω_{PBH}≳10^{-6}Ω_{CDM} of heavy M∼10^{4}-10^{7} M_{⊙} PBHs. The latter could play a role in alleviating some of the shortcomings of the standard cosmological model on subgalactic scales. The scenario might have distinct signatures in ongoing axion searches as well as gravitational wave observatories.