My CX frame (Cross Check) fits the bill for being a fairly decent all-rounder.

I've used 28mm Gatorskins for 2 years, just recently swapped up to a pair of 32mm Pasela TGs because many of my long-distance routes take me over lots of chip-seal or even some light crushed stone/gravel/dirt roads.
I throw a pair of 35mm studs on it for the ice and snow.
I'll even be giving it a true test of all-rounder mettle on a 300k this fall which has 20 miles of loose gravel mountain roads. Might even consider a pair of 40mm tires for that venture!

It is a compromise bike, though. It's not the lightest thing on the street. Nor will it be the fastest, no matter who's powering it. It's OK in the snow, but not awesome like having a pair of 2.2" fatties to help float on the fluffy stuff. It can haul some stuff, but not like having a fully loaded touring bike.
If you need, as I do, to go really far at a modest speed with a limited amount of equipment on roads that don't suck too bad, it's a great bike.