Colours opposite one another on the colour wheel are complementary. As the name implies, they work well with one another in any decor scheme. To find the perfect paint pairing for the wood in your space, simply marry the wood undertone to a complementary shade. So...
Oak's orange undertone tooks fab with blues.
The red in mahogany complements greens well.
The greens in barn board and washed wood stun alongside reds.