You could setup a specific hotkey for this.
In the User Preferences > Input > 3D View add a new operator transform.translate
assign whichever available key combination you want, then in the options below tick Constraint > Y
Whenever you press that key combination it will automatically invoke the move command constrained to the desired axis.
If you don't want a different hotkey you can set this temporarily or permanently in the existing G hotkey.
You may also use the Move tool from the toolbar and use the Transfrom gizmo by hovering the desired axis effectively constraining to that vector.