It's not actually illness. The problem is that the tech that keeps him alive is imperfect, so he suffers an irritating cough. The cough is there because George Lucas came into work with bronchitis while they were finalizing Grievous's design and figuring out how he should sound for the movie, so they recorded George coughing and threw it in, the idea being that it would communicate to the audience very quickly that Grievous is not entirely a robot, since there's no reason a robot would cough: it emphasizes that he's a cyborg, and foreshadows Anakin's own cybernetic makeover. A lot of people believe that the cough comes from Mace Windu crushing Grievous's chest in the 2003 Clone Wars show, and they're partially right: the crushing certainly made the cough much worse, just in time for Revenge of the Sith. The actual explanation there, though, is that Clone Wars was made before all the production details for Grievous were settled on for Revenge of the Sith, and once Grievous's movie voice and cough were settled on, John DiMaggio had already been portraying Grievous in Clone Wars for some time.
In the Star Wars prequels, there is a somewhat-important character introduced named 'General Grievous'. Being lord of a large droid army and having some number of sophisticated manufacturing plants at his disposal, I immediately assumed he was a droid himself, albeit a sophisticated one with odd idiosyncrasies (such as coughing). General Grievous is the supreme leader of The Droid Army, and the secondary antagonist of the Star Wars prequel trilogy, first appearing as one of the two main antagonists (alongside Palpatine) of the 2005 film Star Wars Episode IIIl: Revenge of The Sith. He is also a tyrannical commander of The Confederacy of Independent Systems.
The crush was introduced as a means of bridging the gap. With the later release of The Clone Wars, in an effort to maintain continuity between the new show (in its higher canon tier) and the movie (in the HIGHEST canon tier), Grievous was voiced by Matthew Wood, just like in the film, and kept his cough, although his cough was not nearly as bad as in the film. As such, John DiMaggio's '03 voice for Grievous has not been canon for many years, although the rest of his depiction was (though the Attack on Coruscant has been ambiguously and inconsistently portrayed across multiple sources without a single clear answer to the inconsistencies). Now that only the movie saga and TCW are canon at all, Grievous's cough is eternal. I've got one better for you: Other than minor details about how certain voices sound, there are actually ZERO contradictory elements between the old show and the new show.
The only real work that needs to be done is compressing Seasons One and Two of the old show and the Battle of Hypori episode of Season Three (the one that debuts Grievous and has Anakin's knighting ceremony) into the first couple months of the war, and then TCW picks up where it left off, with the rest of Season Three coming in just before Revenge of the Sith starts. It's been excised from the canon because of reasons, I guess. Who knows what those reasons are? But without the old Clone Wars show, Asajj Ventress notably has no canon introduction, and that show is her only EU intro: everything else she's in just sort of assumes the viewer knows about her. Abrams has some serious shit planned, I think the story group should probably toss the old show back in. There are two. The first was done in 2003, by Cartoon Network, and was given to Genndy Tarkovsky (hope I spelled that right), and was meant to be watched as a microseries, with most episodes only being five minutes long.
The only real purpose of this show was to help bridge the gap in episodes 2 and 3 and help with the setup. The second clone wars was started in 2008, and was done by Lucasfilm itself, although it to was released on Cartoon Network. This is the one that tried to tell it's own stories and created whole new characters for this purpose (Ahsoka Tano being the best example).
Download New General Grievous Before He Was A Robot
In terms of canon, the second show is the only one that is now recognized, due to it still airing at around the time the purchase was made by Disney.