We examine transient responses of velocity fluctuations in inertialess channel flows of viscoelastic fluids. Such fluids have broad applications in modern technology including the design and control of polymer processing operations and the development of strategies to efficiently mix fluids in microfluidic devices. For streamwise-constant three-dimensional fluctuations, we demonstrate analytically the existence of initial conditions that lead to quadratic scaling of the kinetic energy density with the Weissenberg number, We. This illustrates that in strongly elastic channel flows of viscoelastic fluids, velocity fluctuations can exhibit significant transient growth even in the absence of inertia. Furthermore, we show that the fluctuations in streamwise velocity achieve O(We) growth over a time scale O(We) before eventual asymptotic decay. We also demonstrate that the large transient responses originate from the stretching of polymer stress fluctuations by a background shear and draw parallels between streamwise-constant inertial flows of Newtonian fluids and streamwise-constant inertialess flows of viscoelastic fluids.