One of the most well-known applications of geometric algebra in engineering is providing a compact formulation of the kinematics of serial robotic manipulators. However, the use of geometric algebra in the field of robotics is still in its early stages, and there are still several open problems that can be addressed with this elegant and compact formulation. In this context, this work introduces a strategy based on conformal geometric algebra to solve the inverse kinematics problem for a class of 6DOF serial robots without a spherical wrist, for which it is known that the inverse kinematics problem generally does not have an analytical solution. To achieve this, a purely geometric strategy extending already existing contributions for the case where the robot has a spherical wrist is proposed. In particular, a point is assigned to each joint of the robot so that the problem reduces to computing the set of all possible joint positions for a given desired position and orientation of its end-effector. These points are found by defining and manipulating several geometric entities such as lines, planes, and spheres. Finally, validation with a real robot of the considered class is demonstrated both in simulation and experimentation.