In the right setting and style, I love antique furniture that combines wicker, rattan, and bamboo, particularly those pieces that give off that old-time hotel or beach lodge feel. I had a white wicker desk and chair in my room growing up. I'm sure Mom loved it when I broke off little pieces on the back of the chair--either trying to change the design or just not being able to resist testing the strength of the fibers. Naughty.
Above, top to bottom:
~1930s-era wicker and bamboo "steamship" chair with a reclining back and extending footrest from Lennegan & Marantz.
~ c. 1880 Brighton Pavilion bamboo and rattan armchair with "splat back" design from Yale R. Burge Antiques, Inc.
~Mid-century French rattan and bamboo chaise from Bittersweet Interiors.
~Rattan and cane child's chair with wind-up music box ("when the child sits, the music begins"!); France, 1920s. Davis & Taft.
~19th-century bamboo and rattan French bistro set, G4 Decor.
There are many more great finds like this at 1stdibs. And, if you're ever in the West Palm area, be sure to check out Bamboo & Rattan (a vintage Florida furniture superstore!).