Ultrasonic cutting with a straight blade is an advanced method for cutting Nomex honeycomb cores. However, crushing has generally been observed on the machining surface of the honeycomb core during ultrasonic cutting (UC). In this paper, to avoid crushing, a new blade-inclined ultrasonic cutting (BIUC) method is proposed to decrease the geometric interference between the cutting tool and the honeycomb cores. The cutting geometry systems of UC and BIUC with a straight blade are systematically established to study the effect of the cutting geometry on the machining quality. The crushing degrees caused by the major flank under different tool orientations in UC and BIUC are analyzed. The causes of crushing during the machining of the honeycomb core were revealed from the aspects of machining quality, cutting force, and geometric interference between the major flank and the material. Experiments involving the quantitative analysis of the crushing and cutting forces verify that the BIUC method is able to avoid crushing by controlling the cutting angle. This paper provides a new method for solving the crushing problem in machining honeycomb core with a straight blade.