Just putting them in a stable solar orbit wouldn't work?
Nope. Orbits work because they balance the inertia of the orbiting objects against the centripetal force of gravity (or whatever is maintaining the orbit*). Since a ringworld/Dyson sphere will basically cover the entire orbit, it's not maintaining that balance.
Consider a planet in orbit; it's being pulled towards the sun** by gravity; that's the force that causes its path to bend into an orbit (you can compare it to "falling" towards the sun, but never hitting, because its own momentum causes it to miss). Now consider a point on a ringworld: it also experiences gravity pulling it towards the sun. However, for every point on the ringworld, there is a point directly opposite on the other side of the sun which is also being pulled towards the sun, but since they are on opposite sides of the sun, the force acts in the other direction. The result is we have
no net force acting of gravity acting on the ringworld as a whole, since every point is exactly cancelled*** by a different point. So there's nothing to sustain the orbit.
*For instance, you get a similar dynamic if you take a ball on a string and spin it around your head; here the tension of the string is acting the same as gravity would for a planet in orbit.
**Technically the center of mass of the system, but that's essentially the sun.
***I've assumed a circular orbit for simplicity, but making it elliptical makes the math messier without making it any more stable.