Cleaning out six bags? That sounds a bit excessive. Just throw the hops straight in the boil. Beyond that, I strain using a nylon bag with a band on my primary bucket. Even so, straining is not 100% required, it's just a personal preference of mine to reduce trub in the primary, you could just dump it all in the primary with essentially no ill effects.
There is absolutely no problem with throwing your pellets directly into the wort, you will get the best utilization that way, but as discussed the hop particulate will be spread throughout the wort, an nice long 3-4 week primary will drop all of the pellet matter out of the beer without issue.
The one problem this presents is when/if you want to wash your yeast, there will be hop matter in the yeast cake, and it can affect the flavor in your successive beers if you used a large amount of hops.
A good cold break, and whirlpooling for 15+ minutes after the boil will also reduce the amount of hop debris, and of course break material that ends up in the fermenter.
I'm brewing a Belgian Dubbel tomorrow, there isa single hop addition at 90 minutes, and I will be using a muslin bag for the 35g of hops my recipe requires.
Also as discussed, cleaning out the nylon straining bags is a pain in the arse, the convenience of disposable muslin bags can outweigh the hassle of the nylon bags, however they are relatively expensive.
Regardless of your method, if you are using bags, just don't pack them too full, and bear in mind that when the pellet hops break apart, they swell and take up a much larger volume than when you placed them in the bag.
RDWHAHB
Cheers!