Bacterial cellulose (BC) is a biopolymer whose properties have been intensively studied, especially for biomedical applications. Since BC has no antimicrobial activity, it is necessary to use bioactive substances for developing wound healing applications. Another drawback of BC is the loss if its water retention capacity after dehydration. In order to overcome these problems, carboxymethyl cellulose (CMC) and turmeric extract (TE) were selected for the preparation of BC composites. Citric acid (CA) was used as the crosslinking agent. These composites were tested as potential antimicrobial wound dressing materials. TE-loaded BC–CMC composites were characterized in terms of their morphology, crystallinity, and thermal behavior. Swelling tests and curcumin-release kinetic analysis were also performed. All the composites tested had high swelling degrees, which is an advantage for the exudate adsorption from chronic wounds. The antibacterial potential of such composites was tested against Escherichia coli (E. coli), Staphylococcus aureus (S. aureus), and Candida albicans (C. albicans). The in vitro cytotoxicity toward L929 fibroblast cells was studied as well. The obtained results allow us to recommend these composites as good candidates for wound dressing applications.