How about a shade of green like this BM Fernwood?

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I think cool or warm work equally well. hgtv has some designer colors at Home Depot that might make it easier to decide and I think Martha Stewart does some wonderful colors that would blend beautifully with the oak.....check them out and buy some samples and see what appeals to you. You can also google houzz family rooms with oak furniture to get some great paint ideas and see what is trending now. Go for it and find a color you love. Just do two shades lighter on color chip to get balanced intensity.

I think people sometimes make too much of a big deal over cool and warm colors. I don't know what colors are assigned as "country". What is the color that has you worried?

We painted over the stained trim with an off white color. The furniture is oak so we were used to seeing a lot of stained wood, even our switch plate covers and electric outlet covers were oak so this is a big change. We painted the walls Sherman Williams Smoky Blue on 2 walls and SW Diverse Beige on 2 walls. We tried a lot of colors swatches on the wall and liked this combo but now that I see it in the room I'm wondering if we should paint over it and change it to something like Benjamin Moore's Kennebunkport Green. Our kitchen is green so I felt like something different in the family room so that's why I went with the blue and tan. Now I'm questioning my decision.