Before you get all in a huff about randomness, I would like to look at a couple of different types of die-mechanics that I have seen over my limited years of gaming.
Deck - Not to be confused with the 52 Card mechanic, this one is employed by pretty much every collectible card game (CCG) on the market, emulating the most successful card game out there: Magic: The Gathering. Other games out there that employ this mechanic are deck-building games like Acension or Resident Evil. The basic principle is that a player draws a hand of cards from their deck of cards in order to play them on their turn. In Magic, a player typically can only have 4 copies of the same card (unless it's basic land), meaning that their probabilities of drawing the card that they might want will be significantly diminished. This is a highly random game, as the best odds of drawing one card out of a deck - even if there are 4 copies - is 6.7%. This mechanic often compensates for this by allowing players to draw multiple cards at the beginning and during the rest of the game, increasing the probability of drawing that card, or letting players hold onto cards if they draw it too soon. While this seems more random than the dice mechanics, there are often cards to mitigate the random nature a little bit, with abilities like drawing additional cards, searching for specific cards, or playing cards that might already be 'dead'.
These are the main mechanics I have seen out there, but they are far from being the only ones. Other randomness mechanics include the famous D20 system employed by Dungeons and Dragons - which also uses other types of dice (D4, D6, D8, D10, and D12) or the D100 system employed by the Dark Heresy and Fantasy Flight Warhammer 40k role playing systems. The difference is that these mechanics are largely employed by RPGs, which are usually less about competition between the players and more about them cooperating against a common foe.
In part 2 I will look at the non-random mechanics that exist in board games and miniatures games and ponder if it's possible to have a fun game without any such random mechanics while remaining fun and interesting to the players.