Antenna structures for an antenna may be formed from a dielectric carrier with metal structures. The metal structures may be patterned to cover all sides of the dielectric carrier. The dielectric carrier may have a shape with six sides or other shape that creates a three-dimensional layout for the antenna structures. The antenna structures may have a tunable circuit that allows the antenna to be tuned. The tunable circuit may have first and second terminals coupled to one of the sides of the carrier. The metal structures may be configured to form an inverted-F antenna resonating element. Portions of the metal structures may form a first arm for an inverted-F antenna and portions of the metal structures may form a second arm for the inverted-F antenna. The antenna may operate in multiple communications bands. The tunable circuit may tune one band without significantly tuning other bands.