AR action is inherently a dirty action, suppressed or not. It mostly comes down to keeping it clean, which can be difficult or easy, go fast or take half a day, depending on several variable. Shooting it clean essentially boils down to simply keeping it clean. There is no magic solution, but there are things you can do that'll significantly cut down on buildup or make it much easier to clean.
Use cleaner ammo (that goes a looooong way, some ammo is just dirtier than others and in general cheaper ammo is dirtier) clean properly, clean often and use a PRI gasbuster charging handle if they make one for your rifle. Gasbuster charging handles and RTV gasket sealer on charging handles can cut down on gas face, but won't keep the action cleaner. Having a solid cleaning kit and method for cleaning does though. KAC's cleaning kit is a fine example of where to start, it is more detail oriented and has more tools. I bought one when I got my rifle, but you could also just as easily make your own using the list of what's included in it. A 20ga., 9mm and 6.5mm brush for cleaning the bcg really helps but there is NO way around using the dental pick, and the more often the easier it'll be. Let that carbon build up and you'll be breaking tips off the dental picks it'll get so damn hard. A chamber brush and IMO, a chamber mop are necessary, as is the guide rod guide and Dewey type cleaning rod (KAC comes with a CF Tippman cleaning rod now). I also take (~.20"?) nylon coated steel "cable", cut it to length (longer or shorter depending on the rifle) and JB weld a Dewey adapter to one end. This makes for a good pull through compact cleaning rod that'll roll up and fit in almost any cleaning kit. In addition to the nice cleaning kit, I have a smaller one with less in it that I keep in my gear or if I'll be out in the sticks.
They're simply dirty ass actions, especially when suppressed. Keep it lubed and it'll run. Take it down and wipe it down and lube up critical areas in the field when you get chances. Clean it well when you get the chance or at the end of the day or before putting it away, whichever comes first. The more often they are cleaned the less dirty they get; once they accumulate the carbon and shit, it's there and it'll keep building up. A better lube helps too, I like TW25B, the oil and the grease, but use CLP for basic wiping down or in the field.
Lubing properly goes a long way too. Since it's a dirty action, having too much oil can be just as bad as not enough, so learn how to lube and clean an AR properly. Again, KAC has good instructions with the M110/SR25 and cleaning kit, more indepth and specific than what a military FM would show. Believe it or not, most people clean and lube the wrong way. But of course they think they are doing it right. Most people overdo it or put lube in places where it should be lightly lubed or not lubed at all.
A pressure washer with hot water and prior soaking down with Simple Green will completely strip and clean any debris, oil or dirt and drit off that weapon, however, it'll strip away any protection it had too. This won't hurt the weapon to do this every great once in a while or maybe just prior to selling, but know it's not necessary nor is it good for the weapon. Another, less damaging way is to use a solvent cleaning station like you'd find in an auto or machine shop --this is how most weapons got cleaned coming out of the field in NTC for example. But it's not ideal to clean that way all the time for the same reason: it washes away any oil or protection that have built up.
I'm not a big fan of adj. gas blocks, IMO if reliability comes down to an adjustment on the gas block, then I either need to build the rifle different/use different parts, or just go with a different rifle, period. If I went with a gas piston, which will keep your action MUCH cleaner, then it'd be a rifle designed from the factory to have one like the HK AR, not aftermarket parts on just any AR.
All the above goes for AR10's as well as AR15's.