One of my beefs with murder mysteries (and "investigation" adventures in general) is that the DM putting together the adventure knows the capabilities of her players' characters. This can be a good thing, but it also means that she can't truly just lay out a scenario without the nagging realization that one PC can speak to the dead, which affects this part of the scenario, or another is just really well connected to the criminal underworld, which has this effect. And so on.
The obvious solution to this is to have your mystery be written by someone who doesn't know the party layout. Then you get a truly impartial adventure, and if the guy who talks to the dead uses his power now, it at least wasn't subconsciously anticipated.
That means pre-written adventures are good, since most of us can't afford to hire a personal professional writer.
Those are my random thoughts on the subject.