Here's how I do it. As Gator said, if it's not crusty then I'll usually skip the electrolysis, if it is crusty then I'll use the electrolysis just to help remove the crusty stuff. Once I'm down to just the tarnish then I use Tarnex, applying it with a small but fairly stiff artist brush, letting it soak for just a few seconds before washing it off - first with a well soaped up wet rag, then a quick rinse in running freshwater and then dry with a soft rag/towel. I'll repeat this process as many times as it takes. I do well over a hundred pieces a year, some worse then others, and this always works well for me. Chains are usually the hardest - "most time consuming", that and those really crusty rings with settings where the crust and tarnish is behind the setting or deep in the cracks and scroll work, this is where the stiff bristled artist brush comes into play.
PS: I also keep a box of the blue surgical rubber gloves on hand and I use them religiously.