You can constrain an individual axis with the Position constraint. For example you can have an object only constrained on its Y-axis.

1. Create a Position constraint. See Constraining an object’s translation with another’s translation for more information.
2. Double-click the Position constraint in the Scene browser to select it.
3. Open the Properties window. The constraints settings display.
4. Expand Constraint Axes and disable the X, Y, or Z axis for the desired translation effect you want to inhibit.
5. To reactivate the unconstrained axes, activate the option.