Everyone else is doing it, so I thought I'd play along. (One note: It's likely at least a few of the YouTube URLs will go bad, but I won't be tending to them if they do. Sorry.)

1. Russian Ark (2002)




2. Once (2006)




3. Heaven (2002)




4. The Royal Tenenbaums (2001)




5. Man on Wire (2008)




6. Wall-E (2008)




7. The Diving Bell & The Butterfly (2007)




8. The Best of Youth (2005)




9. High Fidelity (2000)




10. Downfall (2005)




11. Where the Wild Things Are (2009)




12. I Heart Huckabees (2004)




13. The Barbarian Invasions (2003)




14. Little Miss Sunshine (2006)




15. The Bourne Supremacy (2004)




16. The New World (2005)




17. Juno (2007)




18. The Darjeeling Limited (2007)




19. There Will Be Blood (2007)




20. Paranoid Park (2007)




21. Gerry (2002)




22. Zidane (2006)




23. The Lives of Others (2007)




24. 4 Months, 3 Weeks, & 2 Days (2008)




25. The Return (2003)




26. Fantastic Mr. Fox (2009)




27. Spellbound (2002)




28. A Serious Man (2009)




29. Away From Her (2006)




30. Milk (2008)




31. Before Sunset (2004)



(1/2/10 Update: So my system failed, and I totally forgot to include these three, all of which I loved: Yi Yi, In the Mood for Love, and Waking Life.)