A quite direct argument is to solve exercise 4.4.17 in page 252 of Thurston's 
book 3-D Geom & Topo - the published one in 1997; in this exercise Thurston 
considered the action of the icosahedral group on the unit circle bundle $UTS^2=RP^3$; 
this is a free action, and the point is to describe a fundamental domain and show that 
the pattern of gluing of its boundary is exactly the same as in the Poincare dodecahedral 
space. When doing this exercise, 
one may want to notice two things: 1. On $S^2$, each of the $5$ angles of a 
spherical pentagon face of a dodecahedron is $120$ degrees, in contrast to 
$108$ degrees for a pentagon on plane - this is useful when you try to create a local coordinate system on the circle bundle $UTS^2$; 2. The 
fundamental domain Thurston created here does not look like a dodecahedron 
in the metric sense - it look like a cylinder over a pentagon base; however 
other fundamental domains will mark a line, with quite a slope, on each 
of the five side faces and therefore each side face splits into two "pentagons"... 
With these in mind, exercise 4.4.17 should be very pleasant to work out.