Opened last July, the Mandarin Oriental, Bodrum (mandarinoriental.com), sits on a prime Turkish Riviera beach between the pine-covered hills just outside of Türkbükü and the preternaturally calm waters of the Aegean Sea. Picturesque olive groves, private gardens, and a series of cantilevered reflecting pools complete the serenely stylish scene at the resort, which features 109 terraced rooms designed by the Italian firm Antonio Citterio Patricia Viel Interiors. Off-white Turkish rugs and other Anatolian decor…