Try to set times aside to purposely wake up naturally.Turn off the alarm clock, keep all people and animals out of your room, wear earplugs if you have to. If this is not possible, at least change your alarm to something less jarring than beeping or a talk show, soft music, something classical, works for me.

If this doesn't help or can't be done, I strongly second what MarineRecon said. Finding and waking up between your REM cycles is a surefire way to remember some dreams.