Why alkali metals have bcc structure?
For the moment I will focus on Na, although I may mention other elements in comparison. 

First, it is true that the STP stable phase of Na is bcc. However, the energy difference between the bcc, fcc, and hcp phases is quite small. As one example, the SGTE database (A.T. Dinsdale, CALPHAD 15(4) 317-425 (1991)) gives the Gibbs free energies (relative to BCC) in Joules/mole as FCC = -50 + 1.3T, and HCP = -104 + 2T. These are really small; normally one would expect kJ/mole or 10's of kJ/mole type differences. Some experiments indicate that Na transforms to hcp below 51K. It is also known that at lower temperatures (35K), Na undergoes a martensitic phase transition. So, bcc is stable at STP, but not very from a big-picture thermodynamics view.

For a more detailed view, lets start with Ashcroft &amp; Mermin's Solid State Physics book. They start their description of the alkali metals (Chapter 15) with


  The alkali metals have singly charged ions (whose core electrons from the tightly bound rare gas configuration and therefore give rise to very low-lying, very narrow, filled, tight-binding bands) outside of which a single conduction electron moves. If we treated the conduction electrons in the metal as completely free, than the Fermi surface would be a sphere of radius $k_{F}$...


Then we get


  De Haas-van Alphen measurements of the Fermi surface confirm this free electron picture to a remarkable degree of precision, especially in Na an K, where deviations in $k_{F}$ from the free electron value are at most a few parts per thousand... Thus, the alkalis furnish a spectacular example of the accuracy of the Sommerfeld free electron model. It would be wrong to conclude from this, however, that the effective crystalline potential is minute in the alkali metals. What it does suggest is that the weak pseudopotential method (Chapter 11) is well suited to describing the conduction electrons in the alkalis. Furthermore, even the pseudopotential need not be minute, for except near the Bragg planes the deviation from free electron behavior occurs only to second order in the perturbing potential...


In other words, they (including Na) really are 'simple' metals, but even simple metals might not be as simple as one might expect.  Note that nowhere is the question of fcc vs hcp vs bcc brought up, but this is a textbook. However, being bcc does not prevent the electrons from being close to free.

Moving on to more recent work on the crystal and electronic structure, I'll turn to On the Constitution of Sodium at Higher Densities (which has Ashcroft as a co-author, likely resulting in the excellent writing as well as the references to Wigner and Seitz' work on sodium from the 1930's). Some quotes relevant to the problem at hand include:


  Under atmospheric conditions, the ground state electronic
  structure of sodium therefore conforms to the classic
  nearly free electron system of Wigner and Seitz ...,
  having a single valence electron and an equilibrating density
  where the core volume is certainly an exceedingly
  small fraction of the cell volume. The pseudopotential
  for sodium at this density is quite well approximated as
  local, and under moderate compression a purely local picture
  also continues to suffice: The primary Fourier component
  of the pseudopotential $V_{G}$ diminishes relative to
  the growing bandwidth. But beyond an approximately
  3-fold compression in volume this simple interpretation
  no longer holds. In Fig. 1 we plot the band structure of
  fcc sodium at $r_{s} = 2.3$ and compare it with that of lithium
  at the same density. The bands near points K, W, and L
  of Na are quite different from those of Li, and, interestingly,
  the Fermi surface does remain nearly spherical in
  Na at these elevated densities. 


and


  We have seen here that at the relevant length scales the electron-ion interaction in sodium is no longer weak under compression ..., and its putative "simple" metallic behavior at low densities appears to be an accident of the relative core and unit cell volumes present at one atmosphere.


Stepping deeper in to the simulations, one might look at First-principles study of the structural properties of alkali metals for pseudopotential calculations of bcc, fcc, and hcp in Li, Na, and K. Another interesting paper might be Structural phase stability in third-period simple metals, where calculations on fcc, hcp, and bcc for Na, Mg, Al, and Si are shown. One point of interest in this paper would be Figure 9, where they show the calculated interatomic pair potentials extracted from their calculations, including showing where the nearest neighbors fall. For Na, while the fcc nearest neighbors are closest to the minimum of the pair potential, the first and second bcc neighbors both are in that first minimum. This gives 14 nearest neighbors for bcc in the big minimum vs only 12 for fcc, yielding the possibility (and reality) that the bcc phase can be slightly more stable than fcc.

So, to wrap up:


Being bcc does not limit the simple nearly-free electron behavior for metals.
For sodium, the energy differences between bcc, fcc, and hcp are really small. The observed stability of bcc at STP is the result of slight differences in the effective pair potential (as well as dynamic effects such as the electron-phonon coupling).
Lower temperature or higher pressure push the balance towards other phases. 


I would be happy to expand discussion on specific points if so desired.