Atomic radius and nuclear effective charge in a group
Just so that we are on the same page before I attempt to answer this question, the definition of effective nuclear charge we are considering Zeff=Z−σ, where Z is the nuclear charge (number of protons) and σ is the shielding factor of the inner shell electrons. The overall effective nuclear charge experienced by the valence electron in question is Zeff.

You are right in thinking that the effective nuclear charge increases down as a group as the increasing nuclear charge has a greater effect on Zeff than the shielding effect of more inner electron shells. This, as you quite rightly mention, should suggest that because Zeff increases down a group, a greater force of attraction would be experienced between the outer shell valence electrons and the nucleus, thus decreasing the atomic radius. However, what we have failed to consider is the size of the electron orbitals in question. The principal quantum number, n, of electron orbitals affects their size. The quantum mechanical description shows that the 1s orbital of a hydrogen atom is a lot smaller in radius than the 2s orbital of a lithium atom for example. This comparison can be further extended where the 3s orbitals are larger in radius than 2s and these are larger than 1s. Thus we have a quantum mechanical picture whereby the principle quantum number of electron orbitals affects their radius. 

Essentially we have two competing factors that affect the trend of atomic radii in question:


The effective nuclear charge, Zeff, increases down a group which draws electrons closer towards the nucleus, decreasing atomic radius.
The principal quantum number, n, of electron orbitals that increases down a group and due to the quantum mechanical nature of electrons, the radius of these electron orbitals increases with increasing n, thus increasing in size as you go down the group.


Simply put, the factor of increasing n down a group is greater than the effect caused by the increasing Zeff, thus causing the atomic radius of atoms to increase down a group in the periodic table.