I regularly use my Tricross off road. The standard tyres won't be much good when the chalk is wet - but then very few tyres will be. It's fine in mud or on wet grass, though.
As for speed compared to a road bike, I have a Tricross and a Secteur and I reckon the difference is around / under 5%. The benefits of the Tricross (it is considerably more stable and, of course, I use it on crappy surfaces without worrying about its fragility) mean that I'd choose that style every time if I could only have one bike. I still find it amusing to get all the funny looks from people when riding off road on a bike with drops.
A brief example from just yesterday afternoon. I got home after about 25 miles on and off road on the Tricross but I wanted to go a bit further, so I headed along the river.
As I left the tarmac , I tucked in behind a couple of chaps on MTB's. One turned off almost immediately but the other set off at a pretty good pace along the river, making the most of his suspension. After about 1/4 mile, though, I realised that the gap between us wasn't changing and about 1/2 mile after that, he stopped "for a drink"
so I passed him.
A couple of miles further along, there was a group of people walking their dog and I stopped for a quick chat. As I set off, though, I heard one of them saying that there was another bike behind me, so I upped my pace a little as there's little opportunity to pass there and I didn't want to hold him up.
Bear in mind that I was probably close to twice this guy's age and weight, and we're riding on mud which has set solid, giving a very rough surface. I've also done 30 miles, at least half of it off road, by this point!
About 200m later, at Boveney Lock, the path splits to give two parallel options, one is a properly surfaced cycle path, the other is the muddy river bank. I head for the river and the other chap, on his MTB, goes for the cycle path. I slowed a little as I was no longer holding him up but, when the two paths joined again about 500m later, I was still in front of him. From that point, though, there was only the one path and it was a nicely surfaced wide cycle path, and he had no hope of sticking with me on the flat.
In summary, I was easily riding the Tricross on hard-packed mud as fast as a guy on an MTB, and he had no hope once the surface got smooth.
I reckon a cross bike is almost as good as an MTB off road (other than on really rough surfaces) but a load better than an MTB on properly surfaced paths and roads. It's the same the other way too. Although I am working harder, I can ride with roadies on the cyclo-cross bike but I'll still be riding when they are carrying their fragility on their shoulders when we run out of tarmac.
) one for yourself first! The style suits me and my requirements, that doesn't mean it will work for everyone. 
. The answer is staring me in the face:
.
