My guess is that old weathered boards have been recycled for generations regardless of rats
I've seen a variety of suggestions for cleaning wood cutting boards that might be appropriate if you have concerns, including wiping with full strength white vinegar, spraying with hydrogen peroxide (the 3% you find at the pharmacy), wiping with a 10% bleach solution, or scrubbing with hot soapy water. Wood is porous, so complete disinfection beyond the surface is likely not possible.
In the mold abatement industry, porous surfaces such as wood beams that have to stay in place for structural reasons are cleaned with an appropriate mold-killing solution, dried, and then encapsulated to seal the surface. You might want to try that approach for the boards. Clean them in your preferred manner, then seal them with a clear varnish or polyurethane finish.