The PCA/covariance/eigenvector method essentially finds the axes of an ellipsoid that approximates the vertices of your object. It should work for random objects, but will give bad results for symmetric objects like the cube. That's because the approximating ellipsoid for a cube is a sphere, and a sphere does not have well defined axes. So you're not getting the standard axes that you expect.
Perhaps if you know in advance that an object is, for example, a cube you can use a specialized method, and use PCA for everything else.
On the other hand, if you want to compute the true OBB there are existing implementations you can use e.g. http://www.geometrictools.com/LibMathematics/Containment/Containment.html
(archived at https://web.archive.org/web/20110817024344/geometrictools.com/LibMathematics/Containment/Containment.html and https://github.com/timprepscius/GeometricTools/blob/master/WildMagic5/LibMathematics/Containment/Wm5ContMinBox3.cpp). I believe this implements the algorithm alluded to in the comments to your question.
Quoting from that page:
The ContMinBox3 files implement an
algorithm for computing the
minimum-volume box containing the
points. This method computes the
convex hull of the points, a convex
polyhedron. The minimum-volume box
either has a face coincident with a
face of the convex polyhedron or has
axis directions given by three
mutually perpendicular edges of the
convex polyhedron. Each face of the
convex polyhedron is processed by
projecting the polyhedron to the plane
of the face, computing the
minimum-area rectangle containing the
projections, and computing the
minimum-length interval containing the
projections onto the perpendicular of
the face. The minimum-area rectangle
and minimum-length interval combine to
form a candidate box. Then all triples
of edges of the convex polyhedron are
processed. If any triple has mutually
perpendicular edges, the smallest box
with axes in the directions of the
edges is computed. Of all these boxes,
the one with the smallest volume is
the minimum-volume box containing the
original point set.
If, as you say, your objects do not have a large number of vertices, the running time should be acceptable.
In a discussion at http://www.gamedev.net/topic/320675-how-to-create-oriented-bounding-box/ the author of the above library casts some more light on the topic:
Gottschalk's approach to OBB construction is to compute a covariance matrix for the point set. The eigenvectors of this matrix are the OBB axes. The average of the points is the OBB center. The OBB is not guaranteed to have the minimum volume of all containing boxes. An OBB tree is built by recursively splitting the triangle mesh whose vertices are the point set. A couple of heuristics are mentioned for the splitting.
The minimum volume box (MVB) containing a point set is the minimum volume box containing the convex hull of the points. The hull is a convex polyhedron. Based on a result of Joe O'Rourke, the MVB is supported by a face of the polyhedron or by three perpendicular edges of the polyhedron. "Supported by a face" means that the MVB has a face coincident with a polyhedron face. "Supported by three perpendicular edges" means that three perpendicular edges of the MVB are coincident with edges of the polyhedron.
As jyk indicates, the implementations of any of these algorithms is not trivial. However, never let that discourage you from trying :) An AABB can be a good fit, but it can also be a very bad fit. Consider a "thin" cylinder with end points at (0,0,0) and (1,1,1) [imagine the cylinder is the line segment connecting the points]. The AABB is 0 <= x <= 1, 0 <= y <= 1, and 0 <= z <= 1, with a volume of 1. The MVB has center (1,1,1)/2, an axis (1,1,1)/sqrt(3), and an extent for this axis of sqrt(3)/2. It also has two additional axes perpendicular to the first axis, but the extents are 0. The volume of this box is 0. If you give the line segment a little thickness, the MVB becomes slightly larger, but still has a volume much smaller than that of the AABB.
Which type of box you choose should depend on your own application's data.
Implementations of all of this are at my www.geometrictools.com website. I use the median-split heuristic for the bounding-volume trees. The MVB construction requires a convex hull finder in 2D, a convex hull finder in 3D, and a method for computing the minimum area box containing a set of planar points--I use the rotating caliper method for this.
