Index cards, as jester said. Basically put some basic stats for each PC and each monster/npc. Put the cards in order of initiative. The DM flips through cards handling each card's turn.
Precalculate Combat modifiers. You should have a section of character sheet (or a whole new sheet) just for combat. It should list each weapon, its stats, and its total attack modifier. If you have a player who stops and adds their strength modifier, level modifier skill modifier, weapon bonus modifier for every attack, they are a moron and should be shot. Stats don't change round to round. If a buff spell gets cast on the PC, recalculate the modifiers for the current weapon in the margin and use those.
Miniatures and a battle mat. By having a gridded mat (make one out of paper if need be and use pennies and Sharpies) a player who isn't paying close attention can snap to the scene on their turn by glancing at the current status on the table, instead of asking for a situation report from the GM.
Get fewer players. Like any game, more than 4 people drags the game down considerably. This is true for any game, even card games.
Limit chit-chat at the table. When people get sidetracked about the latest zombie movie and how that guy with that shotgun was cool, it takes longer to get back to the game. Get it all out of your system before the game. Don't let players say more than 6 words per round (strict), certainly cut them off if they start babbling lengthy instructions to another player.
Read the DMG, it has plenty of tips for speeding up combat, like rolling damage and attacks at the same time.
Try talking to the slowest player. Perhaps they don't realize that they're being pokey. Once they know, they may speed up.
Enforce a time limit for a player to make a decision. Do not let the party strategize for 5 minutes between rounds of combat. Once init is rolled, they should only need a minute to figure out what spell to cast, or to roll attacks. If they don't get started making an action (haven't even rolled their first attack), then they spend the round defending, and move on.
Each player should be paying attention to the action. They should be looking up spells/feats/rules long before their turn. With each guy taking a minute or longer, thats 3+ minutes to look stuff up, which is plenty of time.
Do not have the TV on or other distractions while gaming. That draws player attention, which lags the game.
Janx