Examine your plan. Can you use a single long slash to make two or more simultaneous cuts? For the trickier ones, aim for two solid runs (one to get par, another to collect all of the coins). The game will take the best of both.
It's worth ensuring you fully complete the easier levels, as it will make it a non-issue when you need to skip some of the tough stages later on.
To get a block of ice sliding, you need to cut sharper than a 45 degree angle for it to gain momentum and move away from the rest of the ice mass that it's resting on.
If you see a swingable object attached to a piece of ice, that's always a big clue that you've found an important source of momentum.
You don't have to destroy trolls to keep your Vikings from danger, but you can't let them touch a single one of the snotty blighters.
The captain of your Viking longboat can only chip away at a very small amount of ice in order to free a trapped crew-member. If you have to make a final swipe to bring a Viking on board, make sure you leave him in as small a block as possible, so you don't have to make an annoying restart for the level.
If you're really stuck on a level, pay close attention to its name on the menu screen - this can often give you a simple clue for how to approach the solution. Like, in the level called 'Barrel Roll', you need to rotate the central mechanism entirely. It won't give you the solution, but the level name can very often point you in the right direction.
You can't swipe through the bronze-colored chains that feature in certain levels. When you reach this kind of obstacle, look at the ice surrounding the chain, as you may be able to make a cut that brings the whole thing swinging into a new position.