  It is well known that the Euler method for approximating the solutions of a
random ordinary differential equation (RODE) $\mathrm{d}X_t/\mathrm{d}t = f(t,
X_t, Y_t)$ driven by a stochastic process $\{Y_t\}_t$ with $\theta$-H\"older
sample paths is estimated to be of strong order $\theta$ with respect to the
time step, provided $f=f(t, x, y)$ is sufficiently regular and with suitable
bounds. This order was known to increase to $1$ in some special cases, such as
that of an It\^o diffusion noise, in which case the Euler for RODE can be seen
as a particular case of the Milstein method for an associated augmented system
of stochastic differential equations. Here, it is proved that, in many more
typical cases, further structures on the noise can be exploited so that the
strong convergence is of order 1. More precisely, we prove so for any
semi-martingale noise. This includes It\^o diffusion processes, point-process
noises, transport-type processes with sample paths of bounded variation, and
time-changed Brownian motion. This also yields a direct proof of order 1
convergence for It\^o process noises. The result follows from estimating the
global error as an iterated integral over both large and small mesh scales, and
then by switching the order of integration to move the critical regularity to
the large scale. The work is complemented with numerical simulations
illustrating the strong order 1 convergence in those cases, and with an example
with fractional Brownian motion noise with Hurst parameter $0 < H < 1/2$ for
which the order of convergence is $H + 1/2$, hence lower than the attained
order 1 in the semi-martingale examples above, but still higher than the order
$H$ of convergence expected from previous works.
