If you're going to be riding on rocks and logs, you will need to run more pressure in a cross tire, and yes, grip will suffer.
I've taken mine on various mtb trails. There is one in queens w/ a short bmx style loop, and I was doing the double on my cross bike (and getting some weird looks to say the least).
Then I came up short once, and it almost threw me over the bars. No suspension.
I'd say for single track and tight and twisty you'll be fine. You won't have the gears for steep climbs... but anything too steep to climb w/ your cross gears is faster if you shoulder it and run up anyhow. Again, lots of weird looks, especially if you're passing people left and right.
It won't take the abuse of jumps, smashing through rock gardens or over logs. You will have to learn a soft touch. "Protecting your tires at all times" goes a long way towards the soft touch you are looking for.