We present a dynamical effective medium theory (EMT) of the dielectric properties of nanoparticle aggregates formed from DNA-linked gold nanoparticles. Experimental measurements show that such aggregateshave reduced UV extinction and plasmon bands that are considerably red-shifted and broadened relative to the plasmon absorption feature observed in spectra of dispersed colloid. The EMT, which can be used to reproduce the observed spectral changes, is tested by comparing aggregate spectra calculated using the EMT dielectric function with spectra from explicit coupled particle calculations, and good agreement is found. The EMT dielectric function is used as well in discrete dipole calculations to calculate extinction spectra for a variety of aggregate shapes not amenable to analytic solution, and the sensitivity of the spectra to aggregate shape is examined. We find that the spectra are only weakly sensitive to aggregate shape, and conclude that, when calculating extinction of the DNA-linked aggregates for comparison with experiment, spherical shapes can be assumed.