Remove episode designations! Roman numerals to the back of the line!
The idea is, you watch em in this order, with no episode designations:
A New Hope
Clash of the Jedi
The Empire Strikes Back
Agents of Evil
Return of the Jedi
This would work for the same reason that the 4, 5, 1, 2, 3, 6 order works better (see the thread about that) because it preserves the twists and develops the characters at about the same time they are introduced in the OT.
Proposed crawls:
STAR WARS
Clash of the Jedi
The galaxy is at peace. The wandering Jedi Knights keep order
and justice in the lawless areas.
In the darkness, a threat stirs. Led by the
warrior Darth Vader, the army of the greedy Separatist Confederation
makes an attack on the tranquil world of Alderaan.
Two Jedi, Obi Wan Kenobi and his elder brother Qui Gon, lead the
liberation of the conquered planet...
STAR WARS
Agents of Evil
The Republic's Clone Army is hard pressed to battle the Confederation.
The Jedi are caught in the brutal fighting, spread thin across the galaxy.
The evil Darth Vader has risen again. With the knowledge of the corrupted Jedi,
he has swept into Coruscant and captured its Chancellor.
Above the city planet, Anakin Skywalker and his mentor Obi Wan race to
rescue the beloved leader...
Clash of the Jedi (or a better title) is Episodes I and II.
It starts off with the final battle of TPM, sans Gungans (if permission is obtained, perhaps Lone Gungan Edition's version.) This starts off with Qui Gon talking to the droid about taking the people to Coruscant, but flipped horizontally so he is going towards the end of the hangar where Maul arrives.
Naboo is Alderaan. Obi Wan and Qui Gon (both full Knights) are brothers. After the duel of the fates, Qui Gon's final words are taken as cryptic; Obi Wan only later finds out that a boy single handedly destroyed the control ship. The next scenes show the podrace and the events surrounding it, where Qui Gon went through considerable effort to free the boy from slavery. The camera pulls back, and an older, Ep2 era Obi Wan is watching the footage (recorded by R2 I guess) in the library (just a database he has access to, no Jedi affiliation.) The movie then proceeds as an even shorter Attack of the Phantom-esque cut of Episode 2, which is a pretty good standalone movie when edited correctly. Most of the new cuts would be in scenes with Palpatine where Anakin does not appear, and scenes with the Jedi, who are relatively unimportant. The idea is to keep the focus on the main characters, something the OT does very well.
The major changes are that Anakin and Padme as a couple are not secret, as there's no reason for them to be, and Anakin takes Padme to Tatooine, not because he has a vision of his mother, who isn't a slave, but because he wants to show off his new girlfriend to his mom, and his sister Beru. Only then, he finds out she's kidnapped. The Sand People aren't just some indigenous beast people, they're a failed attempt at making clones by the Confederation, which is why they're humanoid with the weird masks, and they're insane. Anakin has a real motive for revenge on the (real) bad guys, and Padme can be more forgiving - they are pretty close to animals, and they were specifically created to be evil barbarians. Padme is a hawk encouraging a normally peaceful Anakin to war, or at least a supporter of the creation of the army, which explains why she is a target of assassination.
Instead of a large, organized body, the Jedi are roaming warriors who decide what they want to do. The Jedi in the temple under Mace Windu work directly for the Senate. This explains why they don't send more Jedi to go fight Darth Maul, why Mace arrests Palpatine in the name of the senate, etc.
There could also be a subplot similar to Han Solo returning to help Luke blow up the Death Star. The Senate Jedi scold Obi Wan and Anakin for recklessness, but they show up at the end to save them anyway.
Ideally, borrowing Spence's idea, Darth Maul is just Darth Vader, which is an honorific title of the most powerful evil Jedi (no Sith, just evil Jedi.) In the next movie General Grievous is the new Darth Vader reborn, as other people suggested. This is an easy change to make as it can be done in the crawl and most of episode 1 isn't there.
Also as other people suggested, Geonosis and Tatooine are the same. The enemy army is hidden on the other side of the planet, maybe Anakin finds out from a redubbed (car salesman?) C3PO or a dubbed tusken raider.
For simplicity's sake, Boba and Jango Fett are the same, and the Federation and the Separatists are just the Separatist Confederation. Dooku is not named, but called "the Count," and this way, he shows up in the same movie as Qui Gon, which makes him being his master more powerful.
Yoda is not seen until the (Force only!) fight at the end, where the Count and Obi Wan both recognize him. Maybe there's an offscreen line where Obi Wan addresses him as master. He is a mysterious figure so when he comes next in ESB, it's even better.
Agents of Evil is mostly Episode III.
The main changes would be:
Darth Plageuis is Darth Vader instead, a legend.
Anakin keeps his visions about Padme a secret. She dies in childbirth while he is away, relatively early on. Anakin is too ashamed of his dealings to go to the funeral. The deaths of the Jedi feed directly into his ability to keep her alive. Mace Windu's death triggers a scene of her waking up, and the ruminations scene takes place after Anakin marches into the temple (of course he's crying!) This lets her stay alive long enough to be Leia's mom that she remembers. It also explains why she dies while she is still young; Vader no longer cares, and there are no Jedi left to kill to keep her alive. Bail Organa is the king of Alderaan after Padme is seen as dead, making Leia an actual princess.
This is a good video for this, credit to The Cutter:
Padme still alive http://www.youtube.com/watch?v=adxvS_vjOTo&feature=channel_video_title
The Cutter has another good one that's been posted before, where Padme "no longer has any meaning" to Vader http://www.youtube.com/watch?v=5GvpT_7BMAc
Obi Wan and Padme conspire (possible explanation by C3PO, he's not doing anything else on that ship) to fake her continued pregnancy in order to hide the children from Anakin, which is why she's still got the belly on Mustafar, unless someone wanted to do the creepy work of removing it.