In this paper, we propose a new modeling approach and a fast algorithm for 3D motion planning, applicable for fixed-wing unmanned aerial vehicles. The goal is to construct the shortest path connecting given initial and final configurations subject to motion constraints. Our work differs from existing literature in two ways. First, we consider full vehicle orientation using a body-attached frame, which includes roll, pitch, and yaw angles. However, existing work uses only pitch and/or heading angle, which is insufficient to uniquely determine orientation. Second, we use two control inputs to represent bounded pitch and yaw rates, reflecting control by two separate actuators. In contrast, most previous methods rely on a single input, such as path curvature, which is insufficient for accurately modeling the vehicle's kinematics in 3D. We use a rotation minimizing frame to describe the vehicle's configuration and its evolution, and construct paths by concatenating optimal Dubins paths on spherical, cylindrical, or planar surfaces. Numerical simulations show our approach generates feasible paths within 10 seconds on average and yields shorter paths than existing methods in most cases.