I'm a proponent of "free roaming" hops myself. I just chuck the pellets/cones in loose after the krausen has dropped. My dry hops are generally in for 10-15 days and I've never gotten any grassiness. I make sure my beer is at 68F or higher while I'm dry-hopping. I've never done a side-by-side, but I've heard several people say dry-hopping at warmer temps minimizes grassiness.
I have used both whole-cone and pellet hops successfully, and haven't noticed a flavor difference between the two. Because of this, I prefer pellets for dry hopping because you lose less beer. If you use dried whole cones loosely, you may want to allow an extra day or two of contact time, since they tend to float and will take some time to hydrate.
I also make sure to line my bottling bucket with a sanitized paint strainer bag before I rack over. This keeps any hoppy bits from getting in the bottles.
I usually dry-hop at a rate of 2 oz for a Pale Ale or 3-4 oz for an IPA (per 5 gallons). There is definitely a difference between using 1 and 2 oz of dry hops (to my nose at least), but I'm not quite sold yet that there is a huge difference once you go above 2 oz of dry hops.