Traditional initiative is abandoned to compensate for the complexity of additional dice rolling, as well as for streamlining the DM's organizational obligations during play. Instead (and excepting a surprise round if any), combat is resolved starting with the player at the DM's left and continuing clockwise around the table. NPCs act last each round (although acting out-of-turn could be a good vector for unique monster abilities).
Each character has a "combat speed" attribute which begins at 6 and improves slowly by level depending on character class. Each weapon has a "weapon speed" which is defined as a type of die (d4, d6, d8, d10). On each player's turn, they roll their weapon speed die as many times as possible without exceeding their character's combat speed. This determines the number of attacks (to a minimum of one) they are allowed to make with the chosen weapon. A character may forgo the opportunity for additional attacks to instead use their speed die as additional damage on a single attack. As you'll see below, the opportunity to make additional attacks is frequently a currency for purchasing special conditions (fighting defensively, making attacks of opportunity, etc).
The benefit here is that it includes an exciting element of luck, tactical considerations, and meaningful choices for the player which are influenced by their class and weapon choices. The barbarian with the greataxe is encouraged (but not required) to hit slow and heavy. The thief with the dagger is encouraged (but not required) to get in there and shank away (picture the prison assassination scene from Breaking Bad). It's trivial to come up with conditions in which the obvious tendency would be subverted, as in the above barbarian being mobbed by low-hp kobolds, or the thief making the most of his surprise backstab. It also builds additional, but situational and optional, de facto competence into the fighter via the following mechanics, due to his tendency toward slower weapons with higher damage making multiple attacks less likely (and therefore less valuable) at lower levels.
Defensive Fighting: A character may choose to forgo the possibility of additional attacks to instead fight with care for his own defense, gaining a +4 to AC until his next action.
Attacks of Opportunity: A character may choose to forgo the possibility of additional attacks to instead gain a free attack on any and all opponents who enter or exit his threat range (or otherwise open themselves up to attack) until his next action. This allows the fighter to control the battlefield somewhat, preventing opponents from reaching his companions, for example.
Each character has a "movement speed" (default of 6 for humans), which translates into 5' per point of speed. A character chooses at the beginning of their turn whether to move, attack, cast a spell, use an item, etc--but it is possible to move and act at the same time by taking a -2 penalty to Armor Class. Moving in a straight line and attacking qualifies as a charge, granting +2 to hit.
So that's the basic system. In part 2 and beyond we'll cover some non-traditional combat options, such as unarmed fighting, grappling, spell-casting, and non-lethal damage. I'm curious to hear your thoughts so far though.