The system would feature orbits in all three dimensions, not relatively flat like ours is.
Without some form of manual intervention, this is literally impossible. For this to happen, you need some magic or advanced technology to move the planets into the 3d orbits after the solar system has formed.
While the system is being formed, it is a large cloud of particles. The center of mass of this cloud will be where the star eventually forms. Every single particle around this center of mass has some angular momentum in relation to the center.
If you compare all the particles against each other and average out their angular momentum, you will get a single vector that corresponds to the eventual plane of the ecliptic of the system. Take, for example, two particles, one going around the Center of Mass twice an hour, the other going in the opposite direction at half the speed. The angular momentum of the whole system is 1 revolution per hour in the direction of the first particle.
Over time, as these two particles go past each other they will both slow down. The slower one of the two will eventually not have enough speed to stay in orbit. It will fall into the center of mass, eventually creating a star.
The same is true even as you add more particles. The gravitational interactions of all the particles causes a 'drag' that lowers the angular momentum of each particle towards the average angular momentum of the entire system. This causes most particles to fall into the center of mass creating the system's star. Everything that doesn't fall into the star clumps together to form planets and debris (moons, asteroids, comets, etc).
At the scales we are talking about, you can almost treat an accreting solar system as a fluid. If you have water in a bucket and start spinning the water around, the water particles bumping against each other force them all to go the same direction.
Now to directly answer your question: If, against all odds, a planet formed counter/perpendicular to the direction of the rest of the system then it would slowly change orbits until it was either aligned with everything else, crashed into something (most likely the sun), or - most likely - would pass close to another body and get ejected from the system.