This is my first Math Overflow answer, so hopefully people will go easy on me as well. Everything I'm about to say is based on the following paper of Mark Hovey and Keir Lockridge: http://arxiv.org/abs/1001.0902
You are thinking about $\mathcal{D}(R)$, the derived category of $R$. This category is equivalent to $\mathcal{D}(HR)$ where $HR$ is the Eilenberg-Maclane spectrum associated to $R$. This should explain how Ring Spectra get involved. Chain complexes with flat homology are objects $X\in \mathcal{D}(HR)$ with flat dimension zero. According to Proposition 1.4 in the paper, the following are equivalent:
Flat dim $X=0$
In the Universal Coefficient Spectral Sequence
$E_{s,t}^2 = Tor_{s,t}^R(H_*(X),H_*(Y)) \Rightarrow \mathcal{D}(HR)(X,Y)_{t-s}$
we have $E_\infty^{s,*} = 0$ for all $s>n$ and for all objects $Y\in \mathcal{D}(HR)$.
- There is an exact triangle
$A\to X\stackrel{g}{\to}W\to \Sigma A$
with $H_*(A)$ projective and $g$ phantom (see below for a definition).
- Every map $F\to X$ from a compact object $F\in \mathcal{D}(HR)$ factors through a compact $B$ with $H_*(B)$ projective.
Definition: A map is $g:X\to W$ phantom if for all compact $Z\in \mathcal{D}(HR)$, for all $f:Z\to X$ we have $f\circ g = 0$
That's the only other characterization I know. If you can prove more about your ring $R$, e.g. something about its weak dimension, then that could tell you a lot more about the collection of $X$ with $H_*(X)$ flat.