The 3D Sonic games, especially Sonic 2006 and Unleashed, had really awesome FMVs. In Sonic 2006 Shadow pulls a crowning moment of awesome by removing his power-inhibiting rings and blasting through a sea of Mephiles clones. Sonic, on top of that, saves Elise from a crashing Eggman airship by jumping among falling scrap metal, in trademark Sonic style. Sonic Unleashed opens with a 7-minute FMV with Sonic crashing Eggman's fleet, turning Super, and handing Eggman his own ass. Sega also released a nicely-done 11-minute CG short about Sonic and Chip exploring a haunted house (use this link if your computer has trouble viewing the video.)
The games themselves were not masterpieces. Adventure was flawed by a loopy camera and too many characters; Adventure 2 had too little Sonic gameplay; Heroes had unintuitive gameplay with glitches galore; Shadow is a joke overall; Sonic 2006 had clunky gameplay and a storyline resembling bad sci-fi; Unleashed has the Werehog and an aggravating town hub.
I personally enjoyed the anime-style Sonic The Movie, and the Night of the Werehog short film. Now, Sega has an over-insistence on making a story for each game beyond "Eggman's at it again, go beat him up and save flickies!" Sometimes they go overboard with it, as we've seen in Shadow and Sonic 2006. They've also demonstrated a near-Pixar mastery of CG in their FMVs. I believe they should make a Sonic movie instead of a new game.
Sonic is speed. At least a third of the movie should have Sonic blasting across the world in the manner demonstrated in Sonic Unleashed. They could even take the idea of segmenting the scenes in "stages", just like the games, to present new and challenging obstacles for Sonic to cross. The convenient springs could disappear entirely, allowing Sonic to leap great distances using his own judgment.
I think it would be a worthy investment of Sega's resources to follow this line of thought, just to give fans something they can actually be proud of.