I once played in a campaign with 11 players (Yes, that's ELEVEN), and not one of them was a cleric. (3 Monks, 1 ftr/rgr, 1 ftr/rog, 1 ftr/brb, 1 wiz/rog, 2 pal, 1 brd, 1 ftr.) At about 6th level, we picked up 1 NPC druid, and 1 NPC cleric/rogue, and promptly lost them later. We not only succeeded, we prospered, winning most fights by picking our moments, and succeeding through sheer firepower in many cases. (We didn't call ourselves "The Company of Nightfall's Favor" for nothing

Our favored time of attack was when the enemy slept, because it would usually buy us a few rounds of confusion to pick off the opposition.)
3E is designed so that you don't ABSOULTELY have to have clerics. Bards and Druids have sufficient healing, rangers and paladins have access both to healing spells and healing wands and staves, healing potions are more affordable, and in the worst case possible, the staunchest fighter can heal up inside of a couple of weeks, and mage-types healing up faster than that.
However, it does require a rearragement of your combat strategy, and time-critical missions are going to be very dangerous, indeed. If your DM does not restrict magic items below the default levels assumed in the DMG, then it can be done. If you run in a more magic-poor world, the cleric class will leave "a good idea" and approach "necessity."