Can ECDSA signatures be safely made "deterministic"?
There is a draft RFC which describes a way to implement deterministic (EC)DSA (with test vectors). In this draft, both $h(m)$ (the hash of the message) and $x$ are used as input to a deterministic PRNG which uses HMAC (that's HMAC-DRBG as specified by NIST); the PRNG output is used to yield $k$. I am not sure your simple multiplication with $x$ would be enough to guarantee security; ideally, a random oracle should be used, and HMAC-DRBG is the closest to a practical random oracle that I could find.

Note that $k$ must be generated uniformly in the $[1, q-1]$ range (where $q$ is the subgroup order). Any information on $k$, even partial (such as: values between $1$ and $2^{160}-q$ are twice as probable than values between $2^{160}-q$ and $q$), can be exploited by the attacker.

(There will be a new draft version with a few text changes in the draft -- but the same test vectors -- as soon as I find the time to do it.)

Edit: the RFC is now published as RFC 6979.