It can take ALLOT of 22 ammo to foul a barrel, so I trust it's the initial clean you're asking about. (a bit of barrel fouling can actually improve accuracy with a 22)
For the serious cleanings, you owe it to yourself to get a bore guide. I use one now for all my bolt guns. (all gun specific) I also use ProShot rods, jags, brushes, etc. Nordic Marskman (Brad) in the east coast might be one of the best resources/suppliers OF rimfire cleaning gear. All of my cleaning inquiries have gone to him in the last year, and most of them end there too. I've bought 2 rods, 2 bore guides...and a portable/travel ProShot kit too. And a gun.
For initial cleanings (I used this for EVERYTHING up until about a year ago) I use the Patchworm Kit from 20/20 concepts;
http://patchworm.com/patchworm.html Excellent product for the $, and they have a .17 kit too. For tougher 22 jobs, using this kit, you can grab their "super intensive felt"
http://patchworm.com/patches-felts.html They do a great job. I have the kit, a spare cord, the felts, and a .17 cord.