Since I know what the answer would be, I don't ask the Nature Conservancy when I ride across their local lands. But I don't recommend that method for elsewhere...
Be sympathetic and respectful of their land rights, and their concern for setting a precedent. Stress that you are not representing an organization and do not seek permission for others, and that this is a one time crossing. Also show concern for the operations and if there are any areas along the route to avoid, whether to leave gates open or closed, whether you should dismount and walk your bike in the vicinity of livestock, etc.
Here in California private landowners that allowed trail easements across their land later saw additional restrictions on the use of their land imposed by courts. This has had a chilling effect on getting private landowner cooperation for new trails. This is a pervasive fear across the country, particularly in areas of checkerboard ownership. If the landowners say no, this may be the underlying reason, even if they do not openly say it.