More obvious stuff:
- Don't run around low on health, or even half health. Being near a tower is no guarantee of survival.
- Have an escape option. Some champions have built in escapes (Tristana's rocket jump, Shaco's Deceive, etc.) Often you'll want to supplement this with Flash, Ghost, or Cleanse.
- Get health, armor, or magic resist if necessary against particular champions. Even if you aren't a tank, sometimes you'll have to get defensive items.
- Exhaust will screw up many (but not all) assassins via blind. Twitch loses a lot of effectiveness if blinded for 3 seconds of his ultimate.
- Silence can work as well.
- Stuns of any kind can often save you.
For any of those, try to avoid using the skill on something else if you fear an assassin popping out. If you are starting a team battle, try to hold off on your stun until you Twitch breaks out of stealth.
Nearly all assassins have low health. Focus them to death with your team.
If the assassin has a single hit ability that hurts a lot, consider getting Banshee's Veil. This works wonders vs. Evelyn, for example. Avoiding her stun weakens her greatly.