I've found success with two approaches:
- Yoga, whatever flavor or sequence you happen to like. What I did was go to classes several times a week for a month, then practice at home in the morning with intermittent ventures to group classes.
- Tom Kurz' morning stretch and warm-up series, recommended for combat sports and general health. You take a few minutes to rotate all your joints, then get your heart rate up with five minutes of anything (running in place, jogging, shadowboxing, whatever), then do arm and leg swings to get your shoulder and hip joints mobilized for the day. This video shows something roughly similar.