A method of growing bioengineered tissues for use in surgical replacement of damaged anterior cruciate ligaments has been invented. An anterior cruciate ligament is one of two ligaments (the other being the posterior cruciate ligament) that cross in the middle of a knee joint and act to prevent the bones in the knee from sliding forward and backward relative to each other. Anterior cruciate ligaments are frequently torn in sports injuries and traffic accidents, resulting in pain and severe limitations on mobility. By making it possible to grow replacement anterior cruciate ligaments that structurally and functionally resemble natural ones more closely than do totally synthetic replacements, the method could create new opportunities for full or nearly full restoration of functionality in injured knees. The method is also adaptable to the growth of bioengineered replacements for other ligaments (e.g., other knee ligaments as well as those in the hands, wrists, and elbows) and to the production of tissues other than ligaments, including cartilage, bones, muscles, and blood vessels.