Hemp and hemp seed press cake—a by-product of hemp oil production—are high-protein, gluten-free raw materials that are often used to enhance the nutritional value of breads. The addition of hemp materials, however, often negatively impacts the technological parameters of breads. Consequently, this study investigated whether and how much the addition of various by-products of hemp seed press cakes to wheat bread mixtures adversely affects the texture and colour profile. The texture profile and colour were determined using a texture analyser and tristimulus measurements. The particle size of raw materials was also measured. Principal component analysis was then used to visualise the correlation between all measured values as well as nutritional parameters. The results showed that the addition of only 1% of some hemp raw materials caused significant technological changes (p > 0.05). Hemp raw materials increased bread hardness and decreased elasticity. The colour of breads containing 1% hemp was also visibly darker than the reference bread. The addition of more hemp led to further darkening and the deterioration of the technological parameters of the products. Consequently, while various hemp materials have high nutritional value, a balance with sensory properties, e.g., textural and colour, has to be reached.