It depends on the type of chestnut. Liver, deep red, orange, light, flaxen, etc. I ride english, but I ride a light, "dusty" chestnut mare with a flaxen mane and tail. The saddle I ride her in (because it's the one that fits her the best) is black with orange underneath on the panels. Because of this I like her with a black bridle (no white) with a black saddle pad with gold/bronze/yellow trim. If I rode her western, I would use dark dark brown tack on her, with a black saddle pad. For me I would wear mostly black with gold/bronze highlights.
Though this might just be my english background wanting to stick closer to neutrals haha.
Attached is the mare I ride with her tack. (note the saddle pad needs a good wash and has been quite sun bleached)
Edit: Link wouldn't work, attaching an outfit I liked.