Always look ahead for your next move. Remember: if you run out of valid moves, the game is over. So check carefully to see whether your move will open additional match options (good), reduce your available matches (bad), or leave you without any further matches at all (d'oh!).
Also remember that matching a set of gems on the bottom of the screen will definitely affect all the gems above it, so if you can see two valid matches, consider the top one first.
Combos and cascades score extra points! The more combos or cascade matches you can make with a single move, the more the bonus points rack up. Look for moves which will either make multiple matches at once, or cause the gems above to fall down into more ready-made matches.
Matching four gems will create a power gem. Using the power gem in a match will cause it to explode, detonating all nearby gems. The catch, of course, is that the more gems you remove at once, the more difficult it is to predict what the board will look like after the explosion. Be careful not to remove all your match options in one fell swoop! Unused power gems carry over between levels, so if you're not sure whether using one would be a good idea, just hang onto it for later.
Matching five gems will create a hyper cube. Swapping the hyper cube with any other gem will remove all gems of that color from the board. Sounds great, but here are three important tips: