This is what I actually did ...
Juice and zest of 4 limes (because that's what I had)
200g grated ginger
Several dashes of ground cloves (again, because that's what I had)
540g agave syrup (we ran out of sugar earlier in the day)
3 quarts water (I wanted to make it more gingery and I don't have giant pot, so I scaled down the water and sugar)
Brought the mixture to a boil, then simmered for an hour or so.
Cooled in a cold water bath, then added 1 package of red star pasteur champagne yeast (i think i read somewhere that champagne yeast is good for ciders, so it seemed like a reasonable choice -- again, i'm a newbie at this, so suggestions on a better yeast would be highly appreciated)
Left the stuff in the pot overnight, strained with a wire mesh strainer (should I have used cheesecloth?) and put into beer bottles, when i then capped.
Put the bottles in a relatively cool room (60-65F maybe) for about 24hrs, then refrigerated.
When I open the bottles, they seems to be over-carbonated and/or have too many little particles that serve as nucleation sites, so it immediately bubbles over. I probably lose 10-15% of the bottle before i can pour it into a glass. But, the ginger beer tastes great. Nice and gingery!