Coming out of Leicester, take the Great Central cycle path down to Glen Parva, then through either Blaby or Narborough to Cosby, then Broughton Astley, Leire, Ashby Parva. From here on I'm only really familiar with the route into Lutterworth, but to avoid the A426 (which I think is busiest south of Lutterworth) there is a route south from Ashby Parva, crossing the A5 by Magna Park, then through Newbold on Avon into Rugby - google it.
Personally I've never found the northern part of the A426 too bad, though it possibly gets some over-fast driving on some of the bends, but the southern part, crossing the A5 and M1 and into Rugby always seemed to have a lot more traffic.