36 or 39mm?
I agree with you that a brown croc is too dressy... if you are thinking of a "bamboo" style croc. There are other croc straps with a rougher finish that can work IMO.
A tan or brown strap in buffalo or a grained leather would work well. Another more unusual option would be distressed grey/charcoal - the melange of light brown flecks showing through the distressed leather can be quite nice with a black-faced watch.
Yet another option would be to look at the LE Tenzing edition of Explorers - can't recall the exact strap off the top of my head but remember it was nice.
Or...
whiskey cordovan, heh heh. Cordovan darkens with wear so IMO the lighter colours are preferable. The stitching in mine actually sets off the creamy tritium in the dial, which you can't see in the bad flash photo.
I think the E1 can take a strap with no stitching, but I think it looks best with some stitching on the strap, whether it's the 2 long parallel ones or a line across the strap near the lugs.
Hope that helps...