In this paper we study compactifications of heterotic string theory on manifolds satisfying the partial derivative(partial derivative) over bar -lemma. We consider the Strominger system description of the low energy supergravity to first order in alpha' and show that the moduli of such compactifications are subspaces of familiar cohomology groups such as H-1 (TX), H-1 (TXV), H-1 (End(0) (V)) and H-1(End(0) (TX)). These groups encode the complex structure, Kahler moduli, bundle moduli and perturbations of the spin connection respectively in the case of a Calabi-Yau compactification. We investigate the fluctuations of only a subset of the conditions of the Strominger system (expected to correspond physically to F-term constraints in the effective theory). The full physical moduli space is, therefore, given by a further restriction on these degrees of freedom which we discuss but do not explicitly provide. This paper is complementary to a previous tree-level worldsheet analysis of such moduli and agrees with that discussion in the limit of vanishing alpha'. The structure we present can be interpreted in terms of recent work in Atiyah and Courant algebroids, and we conjecture links with aspects of Hitchin's generalized geometry to heterotic moduli.