We propose two designs of birefringent microstructured fibers assuring high geometrical birefringence in the visible range and low confinement losses. An increased birefringence level in the visible range is achieved by breaking the hexagonal symmetry of the structure and shifting selected holes with respect to the nodes of the hexagonal lattice. Such geometry of the microstructured cladding is feasible for polymer fibers fabricated using a preform drilling method. In the first optimized design, two large air-holes adjacent to the core are shifted towards each other to increase modal birefringence. In the second design, high birefringence is achieved by approaching two regions with large holes separated by a row of smaller holes. The simulation results show that in the first fiber with a pitch distance of 3.0 μm, it is possible to obtain a phase birefringence of the fundamental mode greater than 1 × 10−4, while the confinement losses do not exceed 0.06 dB m−1 in the visible range. In the second fiber with the same pitch distance, the confinement losses are smaller than 0.3 dB m−1 in the spectral range of 500–1100 nm, while the phase modal birefringence exceeds 1 × 10−4 in the visible range and reaches 4.0 × 10−4 at 633 nm. We also present the fiber fabricated according to the second design and compare the calculated and the measured spectral dependence of birefringence.