Most of the encounters I run in my campaign
average about 2-3 weeks so far, and without making many radical changes to the mechanics. The most important change, however, was to replace the standard d20 roll with 3d6. This reduced the number of combat rounds wasted due to strings of bad rolls, which were not uncommon for us in the early encounters when we first started many months ago. But the overall results produced better consistency in attack rolls which helped speed up combat without having to nerf the monsters. You just need to be careful about throwing monsters that might be "impossible" to hit because characters can no longer roll higher than 18, and very rarely roll higher than 16. Any DM worth his salt should be able to adjust the monster levels accordingly.
Another change I made, though not nearly as drastic as altering the core mechanic, was to break down the combat round into smaller, more manageable phases. No more than two characters will act before some of the monsters take their actions. This means more updates per combat round, but it is usually much easier to update a couple of character actions and a few monster attacks than keeping up with all possible actions for both sides at once.
Beyond these changes, I spent a lot of time and effort putting together cheat sheets for players, tracking threads, maps, and establishing easy-to-follow formats. What it basically comes down to is taking the extra time and effort in the short run to save you more needless time and effort in the long run.
Since the campaign officially started back in September, one group has completed five combat encounters (including skill challenges and heavy roleplaying along the way) and the other has begun their fourth encounter (also including skill challenges and heavy roleplaying). Full hit points for all monsters. No short cuts. Just food for thought before you decide on something. Good luck with whatever you decide.