Through a coordinate transformation approach, crystal orientation dependences of elastic and piezoelectric properties at room temperature have been investigated in a three-dimensional space for rhombohedral bismuth ferrite (BiFeO₃). Elastic constants (stiffnesses) c11', c12', c13' and piezoelectric constants d15', d31', d33' along arbitrary orientations were obtained based on crystalline asymmetry characteristics of 3m point group BiFeO₃. Parameters along specific orientations obtaining the largest values were presented. The max c11' = 213 × 10⁸ N/m² could be achieved in planes with ϕ = 0° and 90°. The max c12' = c13' = 132.2 × 10⁸ N/m² could be achieved along directions at θ = 13° and θ = 77° inside three mirror planes, respectively. The max d15' = 27.6 × 10-12 C/N and the max d31' = 12.67 × 10-12 C/N could be both obtained along directions at θ = 69° inside mirror planes. The max d33' = 18 × 10-12 C/N could be obtained at θ = 0°, along the spontaneous polarization axis. By adopting optimal directions, the elastic and piezoelectric parameters of BiFeO₃ could be significantly enhanced which shows applications for the growth of BeFeO₃ films with preferred orientations and enhanced properties.