A Killing field is preserved by the Ricci flow. By a theorem of Daskalopoulos, Hamilton and Sesum (arXiv:0902.1158), on a compact surface an ancient (defined for all negative time) solution to the Ricci flow which is not a shrinking soliton is diffeomorphism equivalent to the Fateev-Onofri-Zamolodchikov-King-Rosenau one-parameter family of metrics. These sausage metrics have the form
$$ g(t) = \frac{-4\sinh(2t)(dx^{2} + dy^{2})}{1 + 2\cosh(2t) r^{2} + r^{4}}$$
in which $z = x + iy$ is a standard coordinate on the complement of a point in the complex projective line, and $r = \sqrt{x^{2} + y^{2}}$. There is an obvious rotational symmetry. The scalar curvature $R(t)$ of $g(t)$ satisfies the bounds
$$\frac{-2}{\sinh(2t)} = \min_{S^{2}}R(t) \leq R(t) \leq \max_{S^{2}}R(t) = -2\coth(2t).$$
(The long list of names attached to these metrics is explained as follows. For a rotationally symmetric metric on the sphere the Ricci flow can be rewritten as the logarithmic diffusion equation $u_{t} = (\log u)_{zz}$ (not the same $z$ as above). P. Rosenau and J.R. King independently found the above solution in the context of such diffusion equations. In the Ricci flow literature they are usually called the Rosenau metrics or the King-Rosenau metrics. However, Fateev-Onofri-Zamolodchikov found this metric, which they called the sausage metric, earlier, in the context of studying the renormalization group flow for a two-dimensional sigma model, in their paper Integrable deformations of the O(3) sigma model. The sausage model, Nucl. Phys. B 406 (3), 521-565 (1993). (the Ricci flow is the one-loop renormalization group flow).)
On a compact oriented surface of genus at least 2 there is no non-zero Killing field, as follows, for instance, from the classical Bochner argument. Similarly, on a torus, a non-zero Killing field must be parallel. On $S^{2}$ a Killing field generates an isometric $S^1$ action fixing two distinct points; that is, the metric is rotationally symmetric (a proof can be found in the paper of Chen, Lu, and Tian). What this shows is that in the compact case an answer to your question has to be on the sphere or torus. Since the Ricci flow preserves isometries, it is natural to ask what the are interesting rotationally symmetric Ricci flows (A recent survey is arXiv:1103.4669); the theorem above about ancient solutions is one sort of answer.
The following is added to my original answer in response to comments by the original questioner asking for metrics related to the hyperbolic metric on the disk. Rescaling the sausage metrics $g(t)$ to have constant volume (in this case this means multiplying by a constant multiple of $t^{-1}$), they solve the volume normalized Ricci flow, and by a theorem of Chow-Hamilton as $t\to 0$ these rescaled metrics converge (as here can be checked directly) to a round metric on the sphere. Consider the metrics
$$ \tilde{g}(t) = \frac{-4\sin(2t)(dx^{2} + dy^{2})}{1 + 2\cos(2t)r^{2} + r^{4}}$$
which are rotationally symmetric and solve the Ricci flow for $t \in (-\pi/2, 0)$. (Formally they are related to the sausage metrics by a complex rotation $t \to it$). Their scalar curvatures $R_{\tilde{g}(t)}$ are bounded as follows
$$-2\cot(2t) = \min_{S^{2}}R(t) \leq R(t) \leq \max_{S^{2}}R(t) \leq -2\csc(t2).$$
$R(t)$ is everywhere positive for $t \in (-\pi/4, 0)$, but for $t \in (-\pi/2, -\pi/4)$ it is both positive and negative. More precisely, it is positive in an equatorial band, and negative on the complementary disks. As $t \to -\pi/2$ these disks expand to fill the complement of the equator. The homothetic metrics
$$h(t) = -\frac{1}{\sin(2t)}\tilde{g}(t) = \frac{4(dx^{2} + dy^{2})}{1 + 2\cos(4t)r^{2} + r^{4}}$$
are determined by the normalization $\max_{S^{2}}R_{h(t)} = 2$, and satisfy the lower bound $-2 \leq 2\cos(2t)\leq \min_{S^{2}}R_{h(t)}$. As $t \to 0$ these metrics converge pointwise to the round metric on the sphere of volume $4\pi$, while as $t \to -\pi/2$ they converge pointwise on either of the disks complementary to the equator to the hyperbolic metric $4(1 - r^{2})^{-1}(dx^{2} + dy^{2})$ of scalar curvature $-2$. Thus the family $h(t)$ interpolates between the hyperbolic metric and the round metric. There is a similar rescaling of the sausage metrics which interpolates in a similar way between the flat metric on the punctured plane and the round metric on the sphere. There are similar families of metrics on the torus, though I am not going to write them down here. I came across the metrics $h(t)$ thinking about Einstein-Weyl structures on surfaces (this is explained in the arXiv:1011.5723, although there the relations with the Ricci flow and sausage metrics are not mentioned, as I was not then aware of them). I don't know what the characterization of the $\tilde{g}(t)$ in Ricci flow terms is, though I suspect they've been described somewhere in the literature on 2-d sigma models and RG flows.
