Our sawyer dips the pine timbers in some solution before he ships them to us in the summer months. I'm not sure what the chemical is, but it stops the mould growth for about 6 weeks. It's only a surface treatment, so as soon as the timber is sanded, it is gone.
If the mould starts up, we spray the timbers with 3 parts water, one part bleach. This kills the spores. In really humid weather we have to go through the frame a few times and spray it, then again at the site. We don't oil the timbers, so I'm not sure if it would affect the finish.