We present the first field-theoretic calculations of the contribution made by cosmic strings to the temperature power spectrum of the cosmic microwave background (CMB). Unlike previous work, in which strings were modeled as idealized one-dimensional objects, we evolve the simplest example of an underlying field theory containing local U(1) strings, the Abelian Higgs model. Limitations imposed by finite computational volumes are overcome using the scaling property of string networks and a further extrapolation related to the lessening of the string width in comoving coordinates. The strings and their decay products, which are automatically included in the field theory approach, source metric perturbations via their energy-momentum tensor, the unequal-time correlation functions of which are used as input into the CMB calculation phase. These calculations involve the use of a modified version of CMBEASY, with results provided over the full range of relevant scales. We find that the string tension $mu$ required to normalize to the WMAP 3-year data at multipole $ell = 10$ is $Gmu = [2.04pm0.06 extrm{(stat.)}pm0.12 extrm{(sys.)}] imes 10^{-6}$, where we have quoted statistical and systematic errors separately, and $G$ is Newton's constant. This is a factor 2-3 higher than values in current circulation.