In a leaked document issued to music providers, Radio 1 said that it was looking to take a new direction with its themes and jingles on the programme.
The station said that Grimshaw's breakfast show will sound "young, exciting, big and confident" when it gets underway on September 24.
Radio 1 therefore wants a theme that sounds "slickly produced and original", and features "live instrumentation and an identifying motif", reports Radio Today.
"It needs to stand out and not sound like any previous BBC Radio 1 Breakfast Shows," the document says.
"Please steer clear of sung jingles and variants of Breakfast themes and identities past and present."
But Moyles was criticised in 2006 after it emerged that he was being paid royalties every time he played the jingles.
Radio 1's document sets out that the new theme would be bought on an "all rights basis", meaning the station gains the jingle "in perpetuity and exclusively", with the creator just holding the composer rights.
The document has also revealed that the new programme will be called 'The Radio 1 Breakfast Show with Nick Grimshaw'.
Companies have until tomorrow (August 15) to submit their proposed audio, including a main theme, a remix of the main theme and a suggested voiceover.
One user called Chalks posted: "The new show doesn't know the sound they want, they're just desperate to be the complete opposite from Moyles's imaging.
"Not sure what real instrumentation they are after either - big band or any brass is out (too much like Moyles) and guitars, drums and bass will end up sounding like 90s Brit-pop - probably not cool now."
Another person called Pete Wilson added: "So basically it will sound as boring and as same as the rest of the station? Mark my words... watch the ratings drop faster than a Lead Balloon!
"Incidentally if the first quarter ratings are any better than Mr Moyles then I will eat my hat and realise that the life of Sung Jingles on Radio One is over!"