No necessarily, in a projectile motion, the body is under a uniform downward acceleration and yet not moving in a straight line. This is because of the 2 components of velocity. From this example we can conclude that the motion of a particle is decided by the direction of velocity also.
This can also be prover by the equations of motion, which show that a body under constant acceleration can change the velocity( magnitude /speed or both) , thereby changing the direction of motion of Te body.
V = u+at
Where acceleration is constant.