Among the many different types of wound dressings, nanofiber-based materials produced through electrospinning are claimed to be ideal because of their advantageous intrinsic properties and the feasibility of employing several strategies to load bioactive compounds into their structure. Bioactive compounds with antimicrobial properties have been incorporated into different wound dressings to promote healing as well as prevent and treat bacterial infections. Among these, natural products, such as medicinal plant extracts and essential oils (EOs), have proven particularly attractive thanks to their nontoxic nature, minor side effects, desirable bioactive properties, and favorable effects on the healing process. To this end, the present review provides an exhaustive and up-to-date revision of the most prominent medicinal plant extracts and EOs with antimicrobial properties that have been incorporated into nanofiber-based wound dressings. The most common methods used for incorporating bioactive compounds into electrospun nanofibers include: pre-electrospinning (blend, encapsulation, coaxial, and emulsion electrospinning), post-electrospinning (physical adsorption, chemical immobilization, and layer-by-layer assembly), and nanoparticle loading. Furthermore, a general overview of the benefits of EOs and medicinal plant extracts is presented, describing their intrinsic properties and biotechniques for their incorporation into wound dressings. Finally, the current challenges and safety issues that need to be adequately clarified and addressed are discussed.