triangle-plane tests won't get you far because you'll need to test against an actual face of the box, not the whole plane that face is on.
triangle-quad intersection test can be done with a separating axis test as well (as it works with any two convex polytopes). A separating axis test is a test where you test objects A and B as follows:
for each face fA in A
test with fA.normal;
for each face fB in B
test with fB.normal;
for each edge eA in A
for each edge eB in B
test with cross(eA.direction, eB.direction)
Where each 'test' consists of an 1D overlapping test - project both objects on the normal, giving you a min-max range for each object, and test whether these ranges overlap (if not, this normal is the 'seperating axis' and the objects don't intersect eachother)
Of course, parallel and oposite vectors only have to be tested once. Since a triangle has 1 unique plane and 3 unique edges, and a box has 3 unique planes and 3 unique edges, that leaves you with 1+3+(3*3) = 13 tests. A quad has 1 unique plane and 2 unique edges, so you'll end up with only 1+1+(3*2) = 8 axis tests.