Looking for theorem of form: If [curvature hypotheses, etc] then every class in $\pi_1(M)$ is uniquely represented by a closed geodesic

There are two different versions of affirmative answers to your question. I'll state both of them in the language of "constant speed reparameterized geodesics" having a common domain of fixed length, rather than "unit speed geodesics" which would require using domains of varying lengths.

First, I'll describe a base-pointed version. Choosing a base point $p \in M$ has the desirable effect that the fundamental group $\pi_1(M,p)$ becomes a well-defined object. Just to be precise, an element of $\pi_1(M,p)$ is thus a path homotopy class of paths $\gamma : [0,1] \to M$ with endpoints at $p$, where "path homotopy" means that the endpoints are stationary under the homotopy. In this context, if $M$ is a complete Riemannian manifold of nonpositive section curvature then each element of $\pi_1(M,p)$ has a unique representative $\gamma : [0,1] \to M$ with endpoints at $p$ which is a constant-speed reparameterization of a geodesic.

Second, I'll describe a non base-pointed version, although I will still use base-points to express the result. For each base point $p$, the set of conjugacy classes in $\pi_1(M,p)$ corresponds bijectively to the set of homotopy classes of continuous functions $\sigma : S^1 \to M$ (and the latter set is well-defined independent of the choice of $p$). In this context, if $M$ is a compact Riemannian manifold of negative sectional curvature then every conjugacy class in $\pi_1(M,p)$ is represented by a constant speed reparameterized geodesic map $\sigma : S^1 \to M$ which is unique up to rotational isometries of the domain $S^1$.

Note the differences between these two versions: the first one allows noncompactness and allows for nonpositive sectional curvatures; the second one requires compactness and requires negative sectional curvatures. Also, because of the presence of a base point, the first version gives a unique representative on the nose; whereas the second version is a bit weaker, allowing nonuniqueness up to "change of base point" where the base point is allowed to rotate around the curve.

I don't have a particular reference to recommend, but any textbook on differential geometry which covers the Riemann curvature tensor and discusses geodesics and their conjugate points should have this result.

I did some background reading and understand what's going on a little now. Let's consider the based version of the result as in Lee Mosher's answer. I follow his lead by only considering geodesics which are constant speed and parametrized by the unit inverval.

Theorem: Let $M$ be a complete Riemann manifold with non-positive sectional curvature and choose a basepoint $p \in M$. Then, every class in $\pi_1(M,p)$ is represented by exactly one geodesic.

This result sounds very surprising at first, but becomes less so once one knows about the Cartan-Hadamard theorem which essentially says that if a complete Riemann manifold of non-positive sectional curvature is simply-connected, then it is diffeomorphic to $\mathbb{R}^n$ via any of its exponential maps. This at least sounds plausible, since the non-positive curvature hypothesis should make the geodesics "spread out", but I was still rather amazed to learn this result. In particular, this result gives us:

Corollary of C-H: Each pair of points in a simply-connected, complete Riemann manifold of non-positive sectional curvature is connected by precisely one geodesic.

Using the universal cover, it is not difficult to see this corollary proves the theorem we're after. First, lifting the Riemann structure of $M$ to the universal cover $\widetilde M$ gives us a simply-connected, complete Riemann manifold of non-positive sectional curvature. By lifting, every homotopy class of loops based at $p$ in $M$ is in bijective correspondence with the collection of all paths from $\widetilde p$ to $\widetilde p'$ where $\widetilde p, \widetilde p' \in \widetilde M$ are two lifts of $p$. This correspondence preserves geodesics, so the above corollary gives us the theorem directly.