Depends on which type of bounding box you're talking about.
Axis-aligned bounding cubes are one of the fastest ways to do a rough first-pass collision test, before sending those that pass to a more precise collision check. Edit: This is especially true when you have multiple moving objects that may collide with one another.
Oriented bounding cubes require more, and more complex, operations, and wouldn't usually be my recommendation for a first pass collision check, but could be used for the second pass if your game objects occupy the majority of that cube. Which, in this case, they actually are cubes, so that fits perfectly. This case would be an exception where if you have a manageable number of cubes, this could be the only check you need to run. However, if you find your game loop slowing down as the number of objects increases, you'll probably want to find another method to prune some number of objects from the list before running this type of test. (Assuming it's the collision that's slowing the game down, and not some other method that's also running on those objects each frame.)
Other than AABB's, and because the cubes are static, you could also use some form of collision tree. In 3D space, look at Octrees. If visualizing the octree isn't easy at first, look at 2D Quadtrees and then come back to 3D. For static objects, the greatest improvement in performance requires either building the tree offline and storing it in a file, or increasing your level load time by building it at run-time. This is just something to be aware of, and unlikely to be a deal-breaker.
My recommendation, because of the specific scenario you described, would be to try OBB's, because your objects actually are cubes and thus this will be the most accurate test possible. See if the game can handle it without slowing down. If so, no need for anything else. If not, AABB's are relatively easy to implement, try nesting your collision such that you check against the AABB first and only if you find an AABB collision would you attempt the OBB collision check. If that's still not enough, scrap the AABBs, look into Octrees and run your OBB collision only if you find a hit within the tree.