I don't use any formula to get the right combination of browns and greens. What I do is add stuff as I have it and mix it up. If it's not heating up, it needs more greens, it needs to be turned or it needs more water. If it stinks, it needs more browns and probably needs to be turned. In VA, it may also stink if it gets too wet (never a problem for me).

I don't know what your composter is like, but if it has a door near the bottom, you can take a shovelful from the bottom and put it on top every once in a while to help mix it.

Another thing that can be useful is a bulb auger. These are like giant drill bits that are designed to make it easy to make holes for planting bulbs. They're about 3 feet long and about two inches in diameter (there may be other sizes). I put it in a cordless drill and periodically stir the compost with it.

I agree with bpgreen (seems that I normally do).

In addition- greens that turn brown become brown. Green leaves fresh off a tree or grass just cut lose N to the atmosphere as they dry out and turn brown.

Here is a link to a recent discussion, It contains a link to some earlier discussions. I will quote from the part about greens becoming browns.
Quote:I believe that 2 mechanisms are mineralization and denitrification- although they are normally described in soils or water.

In the first- N forms (simple anions, or from proteins) are transformed into NH4+, and NH3 is lost because it is volatile. Normal thermal/ light induced degradation and bacterial action can cause this to happen (going on memory here).

In denitrification NO3- is directly converted to N2 as part of the nitrogen cycle. You may remember basic fishtank chemistry where NH4+ --> NO2- --> NO3- . I guess this is the opposite of that.

Going on memory there. However- green leaves turning brown, fresh cut green grass drying out and turning brown- both are accompanied by loss of N as a volatile. I'm sure that autocatalytic breakdown of the proteins in the greens would cause loss of a variety of N bearing compounds. Manures and urine often contain urea, which has a very simple pathway to NH3 conversion.

Just a smidge of a qualification for clarity. ;)

I'm wondering if my grass clippings ("greens") become "browns" if they stay on the lawn long enough to dry out and turn, well, brown?

Yes, but we are talking longer than most would leave their grass clippings staying on the lawn before they'd qualify as a brown. Ever seen anyone mow a pasture for hay? If not, come down here and I'll let you mow. ;) The point is, after we mow we wait a couple of days before windrowing, and then once windrowed it lays for a week + or - a day depending on weather before baling. Once baled it is still "green" in color and N rating and "stinky" smell.

Granted your short grass will dry and brown faster than hay but if you are going to count it as a primary source of brown (and there are several better options available) make sure it has really, really dried first or you'll find your self with a stinky problem.