Walnut flooring is a relatively modern flooring material as it is installed to be seen and not covered up with wall-to-wall carpet. This doesn't mean that it only suits modern styles, though, so take inspiration from the colour of the floor to choose a decor style that suits your lifestyle as well as your home.
If your tastes tend towards the homely and handmade, pair your dark walnut floor with simple, natural furniture and antiques to create a country cottage feel. This look works particularly well with worn, aged floor boards. You don't need to match objects too carefully, so source simple Shaker style chairs and vintage tables in different woods, and pair with cross-stitch samplers and patchwork quilts. Objects should have a sense of history so should appear used and loved. Gingham curtains would complete the look.
Minimalist style became common in the 1990s and is still much in evidence today. The idea is to pare down belongings to only the attractive and essential. If the walnut floorboards are in good condition, match them with simple sculptural furniture and white fitted cabinets raised off the floor with a shadow gap, which is a space between the cabinets and the floor that, because of the shadow created, makes the cabinets appear to float. The dark walnut will really enhance the white decor. For a truly minimal look, keep the look clean, white and unadorned so no personal belongings are on show. This look is only for neat and tidy people who don't mind a lot of tidying and cleaning.
Walnut floors would be the ideal starting point for an oriental scheme, especially if they are of a darker colour or if you don't mind staining them. Install paper screen room dividers and low seating with plenty of silk cushions. Choose wallpaper with motifs of cherry blossom or dragons in shades of red or pink, and accessorise with highly lacquered accessories and antique tea chests. Ornaments made from jade would complete the look, and red paper lanterns would look good reflecting off a walnut floor.
A mid-century modern look would ideally suit a walnut floor. Modernist furniture designs made from curved plastic in bright fake colours, like teal blue and lime green, contrast well with the integrity of a walnut floor. Accessorise with vintage textiles with space-age themed designs, rosewood tables with tapering legs and atomic-styled clocks. A throw rug in a shag pile would complete the look.