In what units are coefficients in `survreg`'s `dist=exponential`? In what units are coefficients in survreg's dist=exponential?
I'm getting the following output:
Call:
survreg(formula = Surv(x, delta) ~ z, data = expdatauusi, dist = "exponential")
             Value Std. Error     z        p
(Intercept)  0.876      0.431  2.04 0.041835
z           -2.112      0.639 -3.30 0.000952

Scale fixed at 1 

Exponential distribution
Loglik(model)= -17.6   Loglik(intercept only)= -22.5
    Chisq= 9.65 on 1 degrees of freedom, p= 0.0019 
Number of Newton-Raphson Iterations: 4 
n= 30 

 A: In survival regression, the exponential and more generally, the Weibull distribution, is parametererized in its log-location-scale representation.  When $T\sim\operatorname{Weibull}(\theta,\alpha)$ such that
$$
S_T(t) = P(T>t) = e^{-(t/\theta)^\alpha},
$$
the log of $T$ belongs to the the location-scale family of distributions since
\begin{align}
S_{\ln T}(x)&=P(\ln T > x)\\
&=P(T>e^x) \\
&=e^{-(e^x/\theta)^\alpha} \\
&=e^{-e^\frac{x-\ln\theta}{1/\alpha}}.
\end{align}
From this we also see that $-\ln T$ is Gumbel distributed with location parameter $\mu=\ln\theta$ and scale parameter $\sigma=1/\alpha$.  In the output from survreg the intercept plus z times the regression coefficients for z equals $\mu$ and the Scale parameter equals $\sigma$.
While $\theta$ has the same units as $T$ (say, seconds s), the parameter $\mu=\ln\theta$ is dimensionless although it is not very explicit that what we mean by $\ln\theta$ is really $\ln(\theta/1\text{s})$, such that we always get dimensionless quantities before taking logs.  See related thread at math.stackexchange.
The scale parameter $\sigma$ is clearly also dimensionless (and equal to 1 in the exponential case).
A: Edit to account for @Jarle Tufto's comment.
Survival regression models available through survreg are of location-scale form for some transformation of time (usually the log) and are not parametric proportional hazard models in general .
The exponential and Weibull distributions are exceptions.
Below I show how to transform the parameters from survreg(dist = 'exponential') in the proportional hazards setting to facilitate the interpretation.

In its most common form, the proportional hazards model is written
$$
h(t) = h_0(t) \exp(\beta x)
$$
Assuming an exponential distribution for the event times, $h_0(\cdot)$ is constant in time: 
$$
h_0(t) = \lambda
$$
To obtain estimates for the parameters $\lambda$ and $\beta$ from the R output, you can use the following transformations (only valid for exponential distribution):
$$
\lambda = \exp\Big(-\text{(Intercept)}\Big) 
$$
$$
\beta = - z
$$
