My instinct would be to use ONE copper ground rod, close to the system, with the grounds of the three Romex runs coming together at one and only one physical point, at that ground rod.
As you obviously realize, the upside of dedicated runs that are separate for digital and analog components is reduced cross-coupling of digital noise into the analog components. The downside is possible voltage offsets between the chassis (and consequently the signal grounds) of the different components in the system, particularly at high frequencies (such as typical noise frequencies), due to the inductance in the wiring. If your components are unbalanced, meaning that they use single-ended interconnects, these voltage offsets result in extraneous current flow in the shields of those interconnects, that is in common with the current flow of signal return paths.
So to minimize that downside, you want to have the chassis and consequently the signal grounds of all of the components as electrically common as possible, with as little inductance between them as possible. DC resistance will probably be sufficiently low regardless of what you do; the concern is with inductance, that can dramatically increase ground run impedance at noise frequencies.
I think that using one ground rod as I suggested will accomplish that. As long as the grounds are tied together at only one point, so that no ground loops exist, there is no reason to "keep the grounds separated." And having three separate rods in relatively close proximity doesn't really keep them separate anyway.
Regards,
-- Al
