1. Dice Rolling
Dice to me can make or break the fun factor of a game. If overused, then the game becomes less fun for competitive players. The best example I can think of in which dice are overused is the Risk series of games. Risk, in theory, is a good game. In practice, it comes down to whoever was able to defy the odds for the longest, or whoever rolls the most sixes. This is a problem in this game because it freaking takes forever. Dice rolling, and luck in general is best suited to games that don't require a heavy time commitment. That way, if someone got completely hosed by the dice, they can either play again and try for better rolls, or leave the room to punch a pillow.
2. Card/Tile Drawing
In most games, there will likely be either cards or dice, and the rolls you make or the cards you draw can determine who is in a better situation to win. Cards and tiles are generally less invasive to strategy, as there is almost always a decision to be made with whatever you have drawn, whereas dice are cold and unforgiving. Again, if a game is entirely about the cards drawn, it, at least personally, detracts from the enjoyment I'm going to get out of a game.
3. Turn order
This is something that people rarely think of when sitting down to play a casual game, but can greatly affect one's ability to win a game. If turn order is random, and you sit after someone who makes great decisions, then it's going to be tougher for you to capitalize. This is something I've noticed when playing multiplayer games of Scrabble, among other games. I can do everything in my power to not set up the person after me, but if it's a three player game, I have no control over what the person to my left does. They could perfectly set up the person to my right, and as a result, I could lose every game that way. For this reason, especially if gaming with a group that has varying levels of game experience, that turn order should ideally be randomized. However, this is not often practical, and people seem to have an aversion to leaving their chosen chair behind.
I feel that a good game has about 75-80% strategy or decision-making, and 20-25% luck. As my girlfriend recently pointed out to me, it is important to know that going into a game, anyone has a chance of winning, otherwise - what's the point? I once saw a graph which ranked some games as where they fall in the Luck vs. Strategy realm, I'll try to recreate my own version here:
This is again, incomplete and subject to some opinion, but I tend to gravitate toward the center of the graph as far as games that I find most fun to play. My favorites here are Puerto Rico, Race for the Galaxy, Settlers of Catan, and Scrabble. That said, there are aspects of all of these (and most every existing game ever) that I enjoy and respect the value of.
Also, it is fair to point out that there are plenty of games that require other characteristics to be successful such as speed/dexterity, creativity, relationships etc. These are the Pictionaries, Craniums, and Apples to Apples' of the world, which are all in their own right great games and I thoroughly enjoy as well.
Hopefully this can be used as a tool for discussion and debate, and to define perhaps where your own tastes lie.
Hugs and kisses.
-Andrew