This old house is (50's) is wired with cloth covered 12/2 Romex with no ground and all metal boxes. Can I just run a 12 ga or 14 ga solid wire between boxes and then to the ground bus of my service panel or do I have to replace all the wiring with new Romex?
The rules on retrofitting grounds were greatly relaxed with the 2014 electrical code (NEC 2014). Yes, now you can simply run bare or green wire between all your electrical boxes and back to the panel.
The ground wires can follow any feasible route, they don't need to travel with the conductors.
The ground wire must go back to the same panel as the conductors come out of (that's relevant if you have more than one panel).
The ground wires MUST be thick enough (generally as thick as the conductors) and can be thicker -- for instance a 12 AWG ground wire is not acceptable for a 10 AWG/30A dryer circuit, but it is acceptable for a 14 AWG/15A outlet circuit.
Here's the whopper: Multiple circuits can share grounds. You don't have to home-run the ground all the way back to the panel, you need only reach another grounded point whose pathway is thick enough. So you can daisy-chain your grounds from box to box, as long as all the circuits you are grounding come out of the same panel. Or, your 10 or 6 AWG ground to your range or dryer can be a "backbone" providing grounds to many other receptacles.
All ground splices must be done with the same rules as any other splices: inside a junction box or using some sort of splice listed for use outside of a box.
For details, see the National Electric Code, NFPA 70 (2014) Article 250 — Grounding and Bonding. (NFPA now offers free access online to its codes and standards.)
I believe that it MAY be code-accepted to retrofit a grounding conductor as you describe. It would need to follow the same path as the existing wire. Check with your LOCAL electrical inspector before taking that on faith, though. Depending just how ungrounded your system is, you may also need to install a ground rod (or two, or three) so that your service panel ground bus is actually tied to ground.
Considering what will have to be done to retrofit this wire, I believe that the sensible approach is new wiring (if you were going to rip things that far apart), or GFCI's and the old wiring with code-approved labeling noting the lack of ground on those circuits (if you were not going to rip things that far apart).