I was working on two separate but related problems at the same time.
To solve the issue of the lines between the resource grid cells, I just put a solid layer behind it that was colored to match the middlemost color of the grid (ie. not the darkest, not the lightest, but a shade in between) and it somehow managed to take all the lines out. I'm making 500 of these maps, different versions for all 50 states, and the data is much to complex and cumbersome to try merging polygons in Arc. By trial and error I figured out a workaround that'll have to do.
The other issue was clipping. For some reason, when I export some state outlines from Arc and open them in Illustrator, the states are converted into compound objects, and these objects simply will not clip the data correctly. Since you can't make a clipping mask out of a compound object, and you can't make a compound object into a single object I was stuck. (I'm still in utter disbelief that Illustrator CS2 can't do these extremely basic things.) I did finally figure out, no thanks at all to the help menu, that using "crop" from the pathfinder will achieve what I need, but only after exporting the state as a solid shape with no boundaries as PDF from Arc, opening in Illustrator, rasterizing, using Live Trace to convert it back to a vector with considerably fewer components, using Pathfinder to "add" the state back into (sort of) one shape, then using the "crop" function to clip the solar resource to the state. After days of trial and error this is the first thing I've found that will work.
I'm really just flabbergasted at how difficult this seemingly simple operation turned out to be. Most states export as a single shape from Arc. I don't know why a few won't (though I suspect it has something to do with extremely complex shorelines because that seems to be the common denominator). I'm quite accustomed to Arc's shortcomings but I guess I expected a lot more from Illustrator.
Anyway I wanted to thank all who responded and offered suggestions. This was my first experience posting on CartoTalk.