A blade is only as good as its edge, and a diamond blade's edge is very good indeed. So yes, a diamond blade will cut wood - but not all woods, and not all cuts.
Hardwoods are no problem for a diamond blade, but softwoods can be more of a challenge. The softer the wood, the more likely it is to gum up the blade, which will eventually lead to dulling. And even with hardwoods, there are certain cuts that are more difficult for a diamond blade than others. For example, cross-cutting (across the grain) is generally tougher on blades than rip-cuts (with the grain).
But in general, if you're looking for a fast and effective way to cut wood - especially hardwood - then a diamond blade is definitely your best bet.